Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×#11920982 |
| Fitting | C2×C3×B(4,𝔽2) ⋊ D3 |
| derived | C3×#997 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #10593630 |
| commutator, cosocle, hypocenter | C3×#997 | C23 |
| Frattini | C23 | C22×C3⋊S4 |
| Fitting | C2×C3×B(4,𝔽2) | D3 |
| socle | C22×C3 | #1020⋊C2 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C22 | |
2330 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 2330 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 13 points:
(4,5)(6,11,12,7,9,13,8,10);(7,12,13,10,8,11);(1,2)(3,4,5)(7,10)(8,12)(11,13)72 conjugacy classes — view the full character table.
| maximal subgroup of | #72275082, C2×D3×#73026, #72275114, … |
| maximal normal subgroup of | #72275082, C2×D3×#73026, #72275114 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5)(6,11,12,7,9,13,8,10), (7,12,13,10,8,11), (1,2)(3,4,5)(7,10)(8,12)(11,13));