Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#73028 |
| Fitting | C4×B(4,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C24 | C22×S4 |
| Fitting | C4×B(4,𝔽2) | D3 |
| socle | C22 | C2×#1020⋊C2 |
| prime radical | C2×#73028 | C2 |
| cosocle | C2×#997 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
1125 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1125 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,14,18)(6,16,20)(7,15,9)(10,11,19);(6,10)(7,14)(8,17)(15,18);(5,6)(7,13)(8,16)(9,10)(11,17)(12,14)(15,20)(18,19)68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11714182, C4×#11714174, C4×#11714171 |
| maximal normal subgroup of | C4×#11714182, C4×#11714174, C4×#11714171 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(5,14,18)(6,16,20)(7,15,9)(10,11,19), (6,10)(7,14)(8,17)(15,18), (5,6)(7,13)(8,16)(9,10)(11,17)(12,14)(15,20)(18,19));