Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×C4⋊#5935 |
| Fitting | C2×C4×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#1020 |
| commutator, cosocle | C2×#997 | C23 |
| Frattini | C22 | C22×#1020 |
| Fitting | C2×C4×H(5,𝔽2) | D3 |
| hypercenter | C22×C4 | #1020 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
3767 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 3767 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(8,13,11)(9,14,12);(4,6)(7,11,13,8,10,14,9,12);(3,4,5,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72245288, C2×#72245330, C2×D4×#5935, … |
| maximal normal subgroup of | C2×#72245288, C2×#72245330, C2×D4×#5935, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(8,13,11)(9,14,12), (4,6)(7,11,13,8,10,14,9,12), (3,4,5,6));