Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×(#1020⋊C3)⋊C2 |
| Fitting | C24×C4 ⋊ D32 |
| derived | A42 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (#1020⋊C3)⋊C2 |
| commutator, hypocenter | A42 | C22×C4 |
| Frattini | C2 | C2×(#1020⋊C3)⋊C2 |
| Fitting | C24×C4 | D32 |
| socle | C25 | C2×D32 |
| prime radical | C2×(#1020⋊C3)⋊C2 | C2 |
| cosocle | C2×A42 | C23 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C4 | |
964 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(5,9,8,7)(6,12,11,10);(1,2,3,4)(7,8,9)(10,12,11);(5,6)(7,10)(8,12)(9,11)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#10596770⋊C2, C4×#72215889, C4×AΓL(2,𝔽4) |
| maximal normal subgroup of | C4×#10596770⋊C2 |
| maximal quotient of | C4×#72215889 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,9,8,7)(6,12,11,10), (1,2,3,4)(7,8,9)(10,12,11), (5,6)(7,10)(8,12)(9,11));