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Finite groups of Lie type catalogue

A catalogue of 68 objects with explicit orders, simplicity conditions, and category views.

Core idea

68 separately identified objects and parameterized families. Individual finite groups and constrained families retain distinct entries. Orders are exact integers or formulas; simplicity is stated under each entry's parameter restrictions. An isomorphism is a recorded relationship, not a reason to collapse the entries.

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Finite derived group

ObjectOrderSimplicityKind of entry
60486048simple under the stated constraintsspecified object
360360simple under the stated constraintsspecified object
504504simple under the stated constraintsspecified object
1797120017971200simple under the stated constraintsspecified object

Finite exceptional chevalley group

ObjectOrderSimplicityKind of entry
214841575522005575270400214841575522005575270400simple under the stated constraintsspecified object
q36(q2−1)(q5−1)(q6−1)(q8−1)(q9−1)(q12−1)gcd⁡(3,q−1)\frac{q^{36}(q^{2}-1)(q^{5}-1)(q^{6}-1)(q^{8}-1)(q^{9}-1)(q^{12}-1)}{\gcd(3,q-1)}simple under the stated constraintsparameterized family
79974760420757997591004872626808029184007997476042075799759100487262680802918400simple under the stated constraintsspecified object
q63(q2−1)(q6−1)(q8−1)(q10−1)(q12−1)(q14−1)(q18−1)gcd⁡(2,q−1)\frac{q^{63}(q^{2}-1)(q^{6}-1)(q^{8}-1)(q^{10}-1)(q^{12}-1)(q^{14}-1)(q^{18}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family
337804753143634806261388190614085595079991692242467651576160959909068800000337804753143634806261388190614085595079991692242467651576160959909068800000simple under the stated constraintsspecified object
q120(q2−1)(q8−1)(q12−1)(q14−1)(q18−1)(q20−1)(q24−1)(q30−1)q^{120}(q^{2}-1)(q^{8}-1)(q^{12}-1)(q^{14}-1)(q^{18}-1)(q^{20}-1)(q^{24}-1)(q^{30}-1)simple under the stated constraintsparameterized family
33111266033664003311126603366400simple under the stated constraintsspecified object
q24(q2−1)(q6−1)(q8−1)(q12−1)q^{24}(q^{2}-1)(q^{6}-1)(q^{8}-1)(q^{12}-1)simple under the stated constraintsparameterized family
1209612096not simplespecified object
42456964245696simple under the stated constraintsspecified object
q6(q2−1)(q6−1)q^{6}(q^{2}-1)(q^{6}-1)simple under the stated constraintsparameterized family

Finite graph twisted group

ObjectOrderSimplicityKind of entry
211341312211341312simple under the stated constraintsspecified object
q12(q8+q4+1)(q6−1)(q2−1)q^{12}(q^8+q^4+1)(q^6-1)(q^2-1)simple under the stated constraintsparameterized family
7653247968377485393920076532479683774853939200simple under the stated constraintsspecified object
q36(q12−1)(q9+1)(q8−1)(q6−1)(q5+1)(q2−1)gcd⁡(3,q+1)\frac{q^{36}(q^{12}-1)(q^9+1)(q^8-1)(q^6-1)(q^5+1)(q^2-1)}{\gcd(3,q+1)}simple under the stated constraintsparameterized family

Finite projective orthogonal group

ObjectOrderSimplicityKind of entry
q12∏i=11(q2i−1)gcd⁡(2,q−1)\frac{q^{1^2}\prod_{i=1}^{1}(q^{2i}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family
q22∏i=12(q2i−1)gcd⁡(2,q−1)\frac{q^{2^2}\prod_{i=1}^{2}(q^{2i}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family
q32∏i=13(q2i−1)gcd⁡(2,q−1)\frac{q^{3^2}\prod_{i=1}^{3}(q^{2i}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family
qn(n−1)(qn+1)∏i=1n−1(q2i−1)gcd⁡(4,qn+1)\frac{q^{n(n-1)}(q^{n}+1)\prod_{i=1}^{n-1}(q^{2i}-1)}{\gcd(4,q^{n}+1)}simple under the stated constraintsparameterized family
q2(2−1)(q2+1)∏i=12−1(q2i−1)gcd⁡(4,q2+1)\frac{q^{2(2-1)}(q^{2}+1)\prod_{i=1}^{2-1}(q^{2i}-1)}{\gcd(4,q^{2}+1)}simple under the stated constraintsparameterized family
q3(3−1)(q3+1)∏i=13−1(q2i−1)gcd⁡(4,q3+1)\frac{q^{3(3-1)}(q^{3}+1)\prod_{i=1}^{3-1}(q^{2i}-1)}{\gcd(4,q^{3}+1)}simple under the stated constraintsparameterized family
qn2∏i=1n(q2i−1)gcd⁡(2,q−1)\frac{q^{n^2}\prod_{i=1}^{n}(q^{2i}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family
qn(n−1)(qn−1)∏i=1n−1(q2i−1)gcd⁡(4,qn−1)\frac{q^{n(n-1)}(q^{n}-1)\prod_{i=1}^{n-1}(q^{2i}-1)}{\gcd(4,q^{n}-1)}simple under the stated constraintsparameterized family
q2(2−1)(q2−1)∏i=12−1(q2i−1)gcd⁡(4,q2−1)\frac{q^{2(2-1)}(q^{2}-1)\prod_{i=1}^{2-1}(q^{2i}-1)}{\gcd(4,q^{2}-1)}not simpleparameterized family
q3(3−1)(q3−1)∏i=13−1(q2i−1)gcd⁡(4,q3−1)\frac{q^{3(3-1)}(q^{3}-1)\prod_{i=1}^{3-1}(q^{2i}-1)}{\gcd(4,q^{3}-1)}simple under the stated constraintsparameterized family

Finite suzuki ree group

ObjectOrderSimplicityKind of entry
3594240035942400not simplespecified object
264905352699586176614400264905352699586176614400simple under the stated constraintsspecified object
q12(q6+1)(q4−1)(q3+1)(q−1)q^{12}(q^6+1)(q^4-1)(q^3+1)(q-1)simple under the stated constraintsparameterized family
1007344447210073444472simple under the stated constraintsspecified object
15121512not simplespecified object
q3(q3+1)(q−1)q^3(q^3+1)(q-1)simple under the stated constraintsparameterized family
2020not simplespecified object
2912029120simple under the stated constraintsspecified object
q2(q2+1)(q−1)q^2(q^2+1)(q-1)simple under the stated constraintsparameterized family

Projective special linear group

ObjectOrderSimplicityKind of entry
11not simpleparameterized family
66not simplespecified object
1212not simplespecified object
6060simple under the stated constraintsspecified object
6060simple under the stated constraintsspecified object
168168simple under the stated constraintsspecified object
504504simple under the stated constraintsspecified object
360360simple under the stated constraintsspecified object
q(q2−1)gcd⁡(2,q−1)\frac{q(q^{2}-1)}{\gcd(2,q-1)}parameter-dependentparameterized family
q2(q4−1)gcd⁡(2,q2−1)\frac{q^2(q^4-1)}{\gcd(2,q^2-1)}simple under the stated constraintsparameterized family
168168simple under the stated constraintsspecified object
2016020160simple under the stated constraintsspecified object
q3(q2−1)(q3−1)gcd⁡(3,q−1)\frac{q^{3}(q^{2}-1)(q^{3}-1)}{\gcd(3,q-1)}simple under the stated constraintsparameterized family
2016020160simple under the stated constraintsspecified object
q6(q2−1)(q3−1)(q4−1)gcd⁡(4,q−1)\frac{q^{6}(q^{2}-1)(q^{3}-1)(q^{4}-1)}{\gcd(4,q-1)}simple under the stated constraintsparameterized family
qn(n−1)/2∏i=2n(qi−1)gcd⁡(n,q−1)\frac{q^{n(n-1)/2}\prod_{i=2}^{n}(q^i-1)}{\gcd(n,q-1)}simple under the stated constraintsparameterized family

Projective special unitary group

ObjectOrderSimplicityKind of entry
11not simpleparameterized family
q(q2−1)gcd⁡(2,q+1)\frac{q(q^{2}-1)}{\gcd(2,q+1)}parameter-dependentparameterized family
7272not simplespecified object
60486048simple under the stated constraintsspecified object
q3(q2−1)(q3+1)gcd⁡(3,q+1)\frac{q^{3}(q^{2}-1)(q^{3}+1)}{\gcd(3,q+1)}parameter-dependentparameterized family
2592025920simple under the stated constraintsspecified object
q6(q2−1)(q3+1)(q4−1)gcd⁡(4,q+1)\frac{q^{6}(q^{2}-1)(q^{3}+1)(q^{4}-1)}{\gcd(4,q+1)}simple under the stated constraintsparameterized family
qn(n−1)/2∏i=2n(qi−(−1)i)gcd⁡(n,q+1)\frac{q^{n(n-1)/2}\prod_{i=2}^{n}(q^i-(-1)^i)}{\gcd(n,q+1)}simple under the stated constraintsparameterized family

Projective symplectic group

ObjectOrderSimplicityKind of entry
q(q2−1)gcd⁡(2,q−1)\frac{q(q^{2}-1)}{\gcd(2,q-1)}parameter-dependentparameterized family
qn2∏i=1n(q2i−1)gcd⁡(2,q−1)\frac{q^{n^2}\prod_{i=1}^{n}(q^{2i}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family
720720not simplespecified object
2592025920simple under the stated constraintsspecified object
q4(q2−1)(q4−1)gcd⁡(2,q−1)\frac{q^{4}(q^{2}-1)(q^{4}-1)}{\gcd(2,q-1)}parameter-dependentparameterized family
q9(q2−1)(q4−1)(q6−1)gcd⁡(2,q−1)\frac{q^{9}(q^{2}-1)(q^{4}-1)(q^{6}-1)}{\gcd(2,q-1)}simple under the stated constraintsparameterized family