Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×#72221912 |
| Fitting | C2×#14554331 ⋊ C2 |
| derived | #10624755 ⋊ C22×C3 |
| Sylow | #14554331 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72221912 |
| commutator, cosocle, hypocenter | #10624755 | C22×C3 |
| Frattini | C33 | C2×C3×C3⋊(C3⋊D3) |
| Fitting | C2×#14554331 | C2 |
| socle | C2×C3 | C3×#92338 |
| LCS | G ▷ #10624755 | |
| UCS | C1 ◁ C2 | |
4208 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 4208 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 83 points:
(4,9)(5,14)(6,46,23,19,20,49)(7,8)(10,29)(11,51,35,33,21,70)(12,27)(13,28)(15,38)(16,24)(17,54,50,44,22,75)(18,43)(25,58)(26,47,62,34,52,53)(30,40)(31,39)(32,73,55,66,41,80)(36,59)(37,48,56,45,42,57)(60,76,61,82,64,79)(63,77,71,68,81,74)(65,69,67,83,72,78);(1,2)(3,6,19)(4,11,34)(5,17,45)(7,23,46)(8,20,49)(9,26,33)(10,32,68)(12,21,53)(13,35,47)(14,37,44)(15,41,74)(16,42,75)(18,22,57)(24,50,48)(25,60,78)(27,62,51)(28,52,70)(29,63,66)(30,64,69)(31,65,79)(36,55,77)(38,71,73)(39,61,83)(40,72,76)(43,56,54)(58,67,82)(59,81,80);(3,18,14,8,24,16,7,5,43)(4,15,58,12,36,31,13,10,40)(6,50,42,20,17,56,23,22,37)(9,30,29,28,39,59,27,25,38)(11,41,72,21,55,67,35,32,65)(19,45,54,49,57,44,46,48,75)(26,60,63,52,64,81,62,61,71)(33,83,66,70,78,80,51,69,73)(34,74,79,53,77,76,47,68,82);(3,4,9)(5,15,39)(6,21,52)(7,13,27)(8,12,28)(10,30,24)(11,26,23)(14,31,38)(16,40,29)(17,41,64)(18,36,25)(19,47,51)(20,35,62)(22,55,61)(32,60,50)(33,49,34)(37,67,71)(42,65,63)(43,58,59)(44,76,73)(45,74,78)(46,53,70)(48,68,83)(54,79,80)(56,72,81)(57,77,69)(66,75,82)236 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((4,9)(5,14)(6,46,23,19,20,49)(7,8)(10,29)(11,51,35,33,21,70)(12,27)(13,28)(15,38)(16,24)(17,54,50,44,22,75)(18,43)(25,58)(26,47,62,34,52,53)(30,40)(31,39)(32,73,55,66,41,80)(36,59)(37,48,56,45,42,57)(60,76,61,82,64,79)(63,77,71,68,81,74)(65,69,67,83,72,78), (1,2)(3,6,19)(4,11,34)(5,17,45)(7,23,46)(8,20,49)(9,26,33)(10,32,68)(12,21,53)(13,35,47)(14,37,44)(15,41,74)(16,42,75)(18,22,57)(24,50,48)(25,60,78)(27,62,51)(28,52,70)(29,63,66)(30,64,69)(31,65,79)(36,55,77)(38,71,73)(39,61,83)(40,72,76)(43,56,54)(58,67,82)(59,81,80), (3,18,14,8,24,16,7,5,43)(4,15,58,12,36,31,13,10,40)(6,50,42,20,17,56,23,22,37)(9,30,29,28,39,59,27,25,38)(11,41,72,21,55,67,35,32,65)(19,45,54,49,57,44,46,48,75)(26,60,63,52,64,81,62,61,71)(33,83,66,70,78,80,51,69,73)(34,74,79,53,77,76,47,68,82), (3,4,9)(5,15,39)(6,21,52)(7,13,27)(8,12,28)(10,30,24)(11,26,23)(14,31,38)(16,40,29)(17,41,64)(18,36,25)(19,47,51)(20,35,62)(22,55,61)(32,60,50)(33,49,34)(37,67,71)(42,65,63)(43,58,59)(44,76,73)(45,74,78)(46,53,70)(48,68,83)(54,79,80)(56,72,81)(57,77,69)(66,75,82));