Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C34 |
| direct product | C2×#72221903 |
| Fitting | C2×#14554857 ⋊ C2 |
| derived | C33×C9 ⋊ C22×C32 |
| Sylow | #14554857 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72221903 |
| commutator, cosocle, hypocenter | C33×C9 | C22×C32 |
| Frattini | C33 | C2×C32×C3⋊D3 |
| Fitting | C2×#14554857 | C2 |
| socle | C2×C3 | #13091503 |
| LCS | G ▷ C33×C9 | |
| UCS | C1 ◁ C2 | |
1956 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,3,11)(2,7,24)(4,39,34,15,17,62)(5,14,35,20,12,40)(6,21,45)(8,58,61,28,30,77)(9,27,51,10,25,53)(13,65,19,37,38,44)(16,55,52,31,26,74)(18,36,63,41,43,64)(22,69,70,49,46,80)(23,47,66,33,48,68)(29,76,67,57,59,79)(32,56,73,42,54,75)(50,72,81,60,71,78);(1,32,50,20,8,49,5,16,57)(2,29,15,9,22,44,10,60,18)(3,30,69,14,26,76,12,54,71)(4,31,23,41,28,6,19,42,33)(7,59,43,25,46,39,27,72,37)(11,61,81,40,52,80,35,73,79)(13,55,48,17,58,21,36,56,47)(24,78,63,51,67,62,53,70,65)(34,77,66,64,75,45,38,74,68)(82,83);(3,12,14)(7,25,27)(13,36,17)(24,51,53)(26,54,30)(37,39,43)(45,66,68)(52,73,61)(55,58,56)(67,78,70)(74,77,75)(79,80,81)166 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72221920 |
| maximal normal subgroup of | C2×#72221920 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,11)(2,7,24)(4,39,34,15,17,62)(5,14,35,20,12,40)(6,21,45)(8,58,61,28,30,77)(9,27,51,10,25,53)(13,65,19,37,38,44)(16,55,52,31,26,74)(18,36,63,41,43,64)(22,69,70,49,46,80)(23,47,66,33,48,68)(29,76,67,57,59,79)(32,56,73,42,54,75)(50,72,81,60,71,78), (1,32,50,20,8,49,5,16,57)(2,29,15,9,22,44,10,60,18)(3,30,69,14,26,76,12,54,71)(4,31,23,41,28,6,19,42,33)(7,59,43,25,46,39,27,72,37)(11,61,81,40,52,80,35,73,79)(13,55,48,17,58,21,36,56,47)(24,78,63,51,67,62,53,70,65)(34,77,66,64,75,45,38,74,68)(82,83), (3,12,14)(7,25,27)(13,36,17)(24,51,53)(26,54,30)(37,39,43)(45,66,68)(52,73,61)(55,58,56)(67,78,70)(74,77,75)(79,80,81));