Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#73084 |
| Fitting | C23×#127 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #857⋊C2 |
| commutator, cosocle | C2×C42⋊C3 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | C23×#127 | D3 |
| socle | C25 | C2×S4 |
| hypocenter | C42⋊C3 | C22×D4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C23 | |
6974 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 6974 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 16 points:
(3,4)(5,8,16)(6,10,13)(7,11,12)(9,14,15);(1,2)(5,7)(6,9)(8,11)(10,14);(1,2)(3,4)(6,9)(8,11)(13,15);(5,6,7,9)(8,12,11,16)(10,15,14,13)88 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72224760, C2×#72240899, C22×#11713628, … |
| maximal normal subgroup of | C22×#11713628 |
| maximal quotient of | C2×GL(2,ℤ/8), C2×#72214513 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,8,16)(6,10,13)(7,11,12)(9,14,15), (1,2)(5,7)(6,9)(8,11)(10,14), (1,2)(3,4)(6,9)(8,11)(13,15), (5,6,7,9)(8,12,11,16)(10,15,14,13));