Finite Group DatabasebetaFinite Group Databasebeta| order | 34560 |
| isoclinism | isoclinic class of #72227496 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72209669 |
| commutator, hypocenter | G | C1 |
| Frattini, Fitting, cosocle | C25×C3 | A6 |
| socle | C24×C3 | SL(2,𝔽9) |
| LCS | G | |
| UCS | C1 ◁ C3 | |
239 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 78 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 78 points:
(1,3,2)(4,7,14)(5,9,15)(6,11,17)(8,10,12)(19,21,26)(20,23,31)(22,29,42)(24,25,35)(27,39,52)(28,41,60)(30,45,62)(32,37,54)(33,48,58)(34,50,66)(36,47,64)(38,56,51)(40,59,43)(44,57,72)(46,63,53)(49,55,69)(61,71,67)(65,76,68)(70,78,74)(73,75,77);(1,4,8)(2,5,10)(3,6,12)(7,14,16)(9,15,18)(11,17,13)(19,22,30)(20,24,34)(21,27,40,23,32,47)(25,36,53)(26,31)(28,33)(29,43,51)(37,55,70)(38,52,67,46,54,68)(39,57,73)(41,61,75)(44,49)(45,56,71,50,63,76)(48,65,78)(58,72,66,60,69,62)(59,64)55 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((1,3,2)(4,7,14)(5,9,15)(6,11,17)(8,10,12)(19,21,26)(20,23,31)(22,29,42)(24,25,35)(27,39,52)(28,41,60)(30,45,62)(32,37,54)(33,48,58)(34,50,66)(36,47,64)(38,56,51)(40,59,43)(44,57,72)(46,63,53)(49,55,69)(61,71,67)(65,76,68)(70,78,74)(73,75,77), (1,4,8)(2,5,10)(3,6,12)(7,14,16)(9,15,18)(11,17,13)(19,22,30)(20,24,34)(21,27,40,23,32,47)(25,36,53)(26,31)(28,33)(29,43,51)(37,55,70)(38,52,67,46,54,68)(39,57,73)(41,61,75)(44,49)(45,56,71,50,63,76)(48,65,78)(58,72,66,60,69,62)(59,64));