Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C29 |
| direct product | C22×#11715850 |
| Fitting | C24×H(5,𝔽2) ⋊ D3 |
| derived | C22⋊#997 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #87588 |
| commutator, cosocle, hypocenter | C22⋊#997 | C23 |
| Frattini | C2 | C22×#87588 |
| Fitting | C24×H(5,𝔽2) | D3 |
| socle | C25 | #1020 |
| LCS | G ▷ C22⋊#997 | |
| UCS | C1 ◁ C23 | |
108456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 108456 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(5,6,8,7)(9,10,13,11)(12,14,16,15);(3,4)(6,7,8)(10,13,11)(14,16,15);(1,2)(11,15)(13,16)116 conjugacy classes — view the full character table.
| maximal subgroup of | #72252245, C2×#72233726, #72267097, … |
| maximal normal subgroup of | #72252245, C2×#72233726, #72267097, … |
| maximal quotient of | #72233730, C2×#72215542, C2×#72218487, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,8,7)(9,10,13,11)(12,14,16,15), (3,4)(6,7,8)(10,13,11)(14,16,15), (1,2)(11,15)(13,16));