Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×#72214065 |
| Fitting, Sylow | C32×C32⋊H(3,𝔽3) ⋊ C22 |
| derived | C3×C32⋊H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72214065 |
| commutator, cosocle, hypocenter | C3×C32⋊H(3,𝔽3) | C22×C3 |
| Frattini | C32 | C3×C3⋊D32 |
| Fitting | C32×C32⋊H(3,𝔽3) | C22 |
| socle | C34 | D3×C3⋊D3 |
| LCS | G ▷ C3×C32⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
12374 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 12374 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(4,5,6)(9,13)(11,12)(14,15)(17,20)(18,22)(19,23);(5,6)(8,10)(11,15)(12,14)(17,19)(20,24)(21,23);(7,8,15)(9,12,10)(11,14,13)(16,17,19)(18,21,23)(20,24,22)180 conjugacy classes — view the full character table.
| maximal subgroup of | #72253827 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,6)(9,13)(11,12)(14,15)(17,20)(18,22)(19,23), (5,6)(8,10)(11,15)(12,14)(17,19)(20,24)(21,23), (7,8,15)(9,12,10)(11,14,13)(16,17,19)(18,21,23)(20,24,22));