Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#73027 |
| 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×#73027 | C2 |
| cosocle | C2×#997 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
1187 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1187 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)(5,7,14)(6,11,18)(9,13,19)(10,16,20);(6,10)(7,13)(8,15)(14,19);(5,6)(7,12)(8,11)(9,10)(13,17)(14,18)(15,16)(19,20)68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11714170, C4×#11714184, C4×#11714171, … |
| maximal normal subgroup of | C4×#11714170, C4×#11714184, 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)(5,7,14)(6,11,18)(9,13,19)(10,16,20), (6,10)(7,13)(8,15)(14,19), (5,6)(7,12)(8,11)(9,10)(13,17)(14,18)(15,16)(19,20));