Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216 |
| direct product | C24×GL(2,ℤ/4) |
| Fitting | C28 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C25 | C2×S4 |
| commutator, cosocle | C2×A4 | C26 |
| Frattini | C2 | C25×S4 |
| Fitting | C28 | D3 |
| socle | C27 | C2×D3 |
| hypercenter | C26 | S4 |
| hypocenter | A4 | C24×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C25 ◁ C26 | |
155179 conjugacy classes of subgroups; only the 4494 normal subgroups are shown.
This group has 155179 conjugacy classes of subgroups — too many to draw. Only its 4494 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) = 192 |
A minimal faithful permutation representation — 6 generators on 16 points:
(7,8)(9,10,12);(5,6);(13,14)(15,16);(10,11)(14,16);(3,4);(1,2)224 conjugacy classes — view the full character table.
| maximal subgroup of | #120×GL(2,ℤ/4) |
| maximal normal subgroup of | #120×GL(2,ℤ/4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,8)(9,10,12), (5,6), (13,14)(15,16), (10,11)(14,16), (3,4), (1,2));