Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#73080 |
| derived | C22⋊A4 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C2×C42 |
| Frattini, socle | C24 | C22×S4 |
| Fitting | C2×C4×#120 | D3 |
| prime radical | C22×#120⋊C3 | C22 |
| cosocle | C22×C22⋊A4 | C23 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C4 | |
2456 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 2456 conjugacy classes of subgroups — too many to draw. Only its 116 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:
(5,7,8,6)(10,15,14)(11,16,12);(1,2,3,4)(11,12)(13,15);(5,6,8,7)(9,10)(11,13)(12,15)(14,16)112 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11713201, C42×#1020⋊C2, #72341604, … |
| maximal normal subgroup of | C4×#11713201, C42×#1020⋊C2, #72341604, … |
| maximal quotient of | #72342061 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,7,8,6)(10,15,14)(11,16,12), (1,2,3,4)(11,12)(13,15), (5,6,8,7)(9,10)(11,13)(12,15)(14,16));