Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
| direct product | C2×C4⋊#5934 |
| 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 | |
5212 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 5212 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:
(3,4,5,6)(7,8,12,10,11,13)(9,14);(4,6)(8,14,11,9)(12,13);(1,2)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#5934, C2×#72245160, C2×#72245330, … |
| maximal normal subgroup of | C2×D4×#5934, C2×#72245160, C2×#72245330, … |
| maximal quotient of | C2×#72252244 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(7,8,12,10,11,13)(9,14), (4,6)(8,14,11,9)(12,13), (1,2));