Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×A4×#120⋊C3 |
| Fitting, Sylow | C23×#120 ⋊ C32 |
| derived | C26 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×#120⋊C3 |
| commutator, cosocle, hypocenter | C26 | C22×C32 |
| Frattini | C22 | C22×A42 |
| Fitting | C23×#120 | C32 |
| socle | C25 | C2×C3×A4 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C2 | |
3120 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 3120 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) = 14 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 14 points:
(3,6,4)(7,14,13,8)(9,12,10,11);(1,2)(4,6,5)(9,13,10)(11,14,12)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72233459, C2×#72258610, C2×A4×#5450 |
| maximal normal subgroup of | C2×#72233459, C2×#72258610, C2×A4×#5450 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,4)(7,14,13,8)(9,12,10,11), (1,2)(4,6,5)(9,13,10)(11,14,12));