Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| isoclinism | isoclinic class of #1020⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C4×#1020⋊C2 |
| Fitting | C2×C4×#120 ⋊ D3 |
| derived | C22⋊A4 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C23×C4 |
| Frattini | C23 | C23×S4 |
| Fitting | C2×C4×#120 | D3 |
| socle | C24 | C22×S4 |
| prime radical | C22×#1020⋊C2 | C2 |
| cosocle | C2×C22⋊A4 | C24 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C4 | |
5704 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 5704 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(3,4,5,6)(7,13,9,11,14,8)(10,12);(9,10)(11,13);(1,2);(7,8)(9,11)(10,13)(12,14)112 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(7,13,9,11,14,8)(10,12), (9,10)(11,13), (1,2), (7,8)(9,11)(10,13)(12,14));