Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| direct product | S4×M64 |
| Fitting | C22×M64 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×S4 |
| commutator | C2×A4 | C22×C16 |
| Fitting | C22×M64 | D3 |
| socle | C23 | C2×D3×C16 |
| prime radical | C22×S4 | C16 |
| cosocle | A4×C16 | C23 |
| hypercenter | M64 | S4 |
| hypocenter | A4 | C2×M64 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C16 ◁ M64 | |
501 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,4,3)(6,13)(12,16)(14,26)(15,27)(24,29)(25,30)(28,35)(34,36);(2,4)(5,7,11,10)(6,12,13,16)(8,17,21,20)(9,18,23,22)(14,24,26,29)(15,25,27,30)(19,31,33,32)(28,34,35,36);(2,4)(5,6,8,14,9,15,19,28,10,16,20,29,22,30,32,36,11,13,21,26,23,27,33,35,7,12,17,24,18,25,31,34)200 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#1398, S4×#2263, S4×#1419, … |
| maximal normal subgroup of | S4×#1398, S4×#2263, S4×#1419, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,13)(12,16)(14,26)(15,27)(24,29)(25,30)(28,35)(34,36), (2,4)(5,7,11,10)(6,12,13,16)(8,17,21,20)(9,18,23,22)(14,24,26,29)(15,25,27,30)(19,31,33,32)(28,34,35,36), (2,4)(5,6,8,14,9,15,19,28,10,16,20,29,22,30,32,36,11,13,21,26,23,27,33,35,7,12,17,24,18,25,31,34));