Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C22×#73080 |
| derived | C22⋊A4 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C23×C4 |
| Frattini | C23 | C23×S4 |
| Fitting | C23×#120 | D3 |
| socle | C25 | C2×S4 |
| prime radical | C23×#120⋊C3 | C2 |
| cosocle | C2×C22⋊A4 | C24 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C23 | |
8524 conjugacy classes of subgroups; only the 268 normal subgroups are shown.
This group has 8524 conjugacy classes of subgroups — too many to draw. Only its 268 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,6,7,8)(10,13,16,14)(11,12);(3,4)(11,15,12)(13,14,16);(9,10)(11,14)(12,13)(15,16);(1,2)112 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C4×#1020⋊C2, #72341604, D4×#73080, … |
| maximal normal subgroup of | C22×C4×#1020⋊C2, #72341604, D4×#73080, … |
| maximal quotient of | #72341428, D4×#73080, C2×#72342679, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,7,8)(10,13,16,14)(11,12), (3,4)(11,15,12)(13,14,16), (9,10)(11,14)(12,13)(15,16), (1,2));