Finite Group DatabasebetaFinite Group Databasebeta| order | 40320 |
| isoclinism | isoclinic class of #72227519 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2×C3×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | PSL(3,𝔽4) |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
243 conjugacy classes of subgroups; only the 3 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 112 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 112 points:
(1,2)(3,5)(4,6)(7,9)(8,11)(10,14)(12,17)(13,19)(15,22)(16,23)(18,26)(20,28)(21,30)(24,34)(25,36)(27,39)(29,41)(31,37)(32,44)(33,46)(35,48)(38,51)(40,47)(42,55)(43,57)(45,59)(49,62)(50,64)(52,66)(53,67)(54,69)(56,71)(58,74)(60,77)(61,79)(63,81)(65,84)(68,88)(70,80)(72,89)(73,90)(75,85)(76,92)(78,94)(82,95)(83,96)(86,98)(87,93)(91,101)(97,104)(99,105)(100,103)(102,108)(106,111)(107,110)(109,112);(1,3)(2,4)(5,7,10,15)(6,8,12,18)(9,13,20,29)(11,16,24,35)(14,21,31,43)(17,25,37,50)(19,27)(22,32,45,39)(23,33)(26,38,52,46)(28,40,53,68)(30,42,56,72)(34,47,60,78)(36,49,63,82)(41,54,70,71)(44,58,75,69)(48,61,80,81)(51,65,85,79)(55,62)(57,73,84,97)(59,76,77,93)(64,83,74,91)(66,86,67,87)(88,99,106,98)(89,90,100,107)(92,94,102,109)(95,96,103,110)(111,112)18 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,5)(4,6)(7,9)(8,11)(10,14)(12,17)(13,19)(15,22)(16,23)(18,26)(20,28)(21,30)(24,34)(25,36)(27,39)(29,41)(31,37)(32,44)(33,46)(35,48)(38,51)(40,47)(42,55)(43,57)(45,59)(49,62)(50,64)(52,66)(53,67)(54,69)(56,71)(58,74)(60,77)(61,79)(63,81)(65,84)(68,88)(70,80)(72,89)(73,90)(75,85)(76,92)(78,94)(82,95)(83,96)(86,98)(87,93)(91,101)(97,104)(99,105)(100,103)(102,108)(106,111)(107,110)(109,112), (1,3)(2,4)(5,7,10,15)(6,8,12,18)(9,13,20,29)(11,16,24,35)(14,21,31,43)(17,25,37,50)(19,27)(22,32,45,39)(23,33)(26,38,52,46)(28,40,53,68)(30,42,56,72)(34,47,60,78)(36,49,63,82)(41,54,70,71)(44,58,75,69)(48,61,80,81)(51,65,85,79)(55,62)(57,73,84,97)(59,76,77,93)(64,83,74,91)(66,86,67,87)(88,99,106,98)(89,90,100,107)(92,94,102,109)(95,96,103,110)(111,112));