Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C22×#1020⋊C4 |
| Fitting | C22×#379 ⋊ D3 |
| derived | C22⋊A4 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C23×C4 |
| Frattini | C23 | C23×S4 |
| Fitting | C22×#379 | D3 |
| socle | C25 | C2×S4 |
| prime radical | C23×#1020 | C2 |
| cosocle | C2×C22⋊A4 | C24 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C23 | |
10132 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 10132 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 48 |
A minimal faithful permutation representation — 4 generators on 16 points:
(3,4)(10,14,13)(11,12,15);(5,6,7,8)(9,10)(11,13)(12,14)(15,16);(12,15)(14,16);(1,2)112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72246282, C2×D3×#1020⋊C4, C22×#11713200, … |
| maximal normal subgroup of | C22×#11713200, C22×C4×#1020⋊C2, #72341296, … |
| maximal quotient of | C2×D3×#1020⋊C4, #72341289, C2×#72215370, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(10,14,13)(11,12,15), (5,6,7,8)(9,10)(11,13)(12,14)(15,16), (12,15)(14,16), (1,2));