Finite Group DatabasebetaFinite Group Databasebeta| order | 34560 |
| isoclinism | isoclinic class of #72227477 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | #72209614 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C25×C3 | A6 |
| LCS | G | |
| UCS | C1 ◁ C2×C3 | |
831 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 831 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) = 30 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2)(3,10,12,11)(4,5)(6,9,7,8)(13,21,18,22,14,23,16,24,15,19,17,20)(25,26,30,29,28,27);(1,6,9,7,3)(4,10,12,11,8)(16,25,23,20,29)(17,28,19,22,26)(18,30,21,24,27)61 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)(3,10,12,11)(4,5)(6,9,7,8)(13,21,18,22,14,23,16,24,15,19,17,20)(25,26,30,29,28,27), (1,6,9,7,3)(4,10,12,11,8)(16,25,23,20,29)(17,28,19,22,26)(18,30,21,24,27));