Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C2×C35 |
| direct product | C2×C3×#13091108 |
| Fitting | C2×C3×UT(4,𝔽3) ⋊ C2 |
| derived | C34 ⋊ C22×C33 |
| Sylow | C3×UT(4,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #13091108 |
| commutator, cosocle, hypocenter | C34 | C22×C33 |
| Frattini | C33 | C2×C33×D3 |
| Fitting | C2×C3×UT(4,𝔽3) | C2 |
| socle | C2×C32 | #92258 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2×C3 | |
4192 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 4192 conjugacy classes of subgroups — too many to draw. Only its 252 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 32 points:
(4,6,12)(5,9,18)(7,14,22,15,13,23)(8,16,24)(10,20,27,11,19,28)(17,25,29,21,26,30);(4,5,8)(6,19,25)(7,11,17)(9,16,13)(10,21,15)(12,27,24)(14,20,26)(18,29,22)(23,28,30)(31,32);(1,2,3)252 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72221920, C2×#72221898 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6,12)(5,9,18)(7,14,22,15,13,23)(8,16,24)(10,20,27,11,19,28)(17,25,29,21,26,30), (4,5,8)(6,19,25)(7,11,17)(9,16,13)(10,21,15)(12,27,24)(14,20,26)(18,29,22)(23,28,30)(31,32), (1,2,3));