Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×#72221918 |
| Fitting | C2×#14554343 ⋊ C2 |
| derived | C32×C9 ⋊ C22×C33 |
| Sylow | #14554343 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72221918 |
| commutator, cosocle, hypocenter | C32×C9 | C22×C33 |
| Frattini | C33 | C2×C33×D3 |
| Fitting | C2×#14554343 | C2 |
| socle | C2×C3 | C3×#92258 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
3344 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 3344 conjugacy classes of subgroups — too many to draw. Only its 250 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 — 3 generators on 83 points:
(2,21,33,6,9,34)(3,43,13,12,14,42)(4,50,52,16,17,51)(5,18)(7,54,65,19,26,58)(8,57,35,20,29,61)(10,62,66,24,37,81)(11,76,25,39,40,53)(15,48)(22,64,56,31,60,69)(23,63,59,41,30,32)(27,72,46,67,68,36)(28,74,71,55,70,38)(44,75,73,49,77,80)(45,78)(47,79);(1,7,22,18,2,55,5,24,30)(3,46,53,14,25,36,13,8,20)(4,27,61,52,11,76,17,29,72)(6,43,28,19,12,31,10,42,63)(9,38,77,62,23,73,26,56,44)(15,70,66,47,64,34,45,32,58)(16,69,21,50,41,54,51,71,37)(33,60,80,81,74,75,65,59,49)(35,57,78,68,39,79,40,67,48)(82,83);(2,7,24)(6,19,10)(8,25,46)(9,26,62)(11,27,29)(20,53,36)(21,54,37)(22,55,30)(23,56,38)(28,63,31)(32,64,70)(33,65,81)(34,58,66)(35,40,68)(39,67,57)(41,69,71)(59,60,74)(61,76,72)236 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72221920 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,21,33,6,9,34)(3,43,13,12,14,42)(4,50,52,16,17,51)(5,18)(7,54,65,19,26,58)(8,57,35,20,29,61)(10,62,66,24,37,81)(11,76,25,39,40,53)(15,48)(22,64,56,31,60,69)(23,63,59,41,30,32)(27,72,46,67,68,36)(28,74,71,55,70,38)(44,75,73,49,77,80)(45,78)(47,79), (1,7,22,18,2,55,5,24,30)(3,46,53,14,25,36,13,8,20)(4,27,61,52,11,76,17,29,72)(6,43,28,19,12,31,10,42,63)(9,38,77,62,23,73,26,56,44)(15,70,66,47,64,34,45,32,58)(16,69,21,50,41,54,51,71,37)(33,60,80,81,74,75,65,59,49)(35,57,78,68,39,79,40,67,48)(82,83), (2,7,24)(6,19,10)(8,25,46)(9,26,62)(11,27,29)(20,53,36)(21,54,37)(22,55,30)(23,56,38)(28,63,31)(32,64,70)(33,65,81)(34,58,66)(35,40,68)(39,67,57)(41,69,71)(59,60,74)(61,76,72));