Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×#72215889 |
| Fitting, socle | C27 ⋊ D32 |
| derived | #10596782 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72215889 |
| commutator, cosocle, hypocenter | #10596782 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C27 | D32 |
| LCS | G ▷ #10596782 | |
| UCS | C1 ◁ C2 | |
6361 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 6361 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4,5)(7,10,9)(8,11)(12,13);(4,6)(7,8)(9,13)(10,11)(12,14);(7,8)(9,12)(10,11)(13,14)60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215896, C4×#72215889 |
| maximal normal subgroup of | C4×#72215889 |
| maximal quotient of | C2×#72215896, C4×#72215889 |
| outer automorphism group of | C22×C3×Q8 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,5)(7,10,9)(8,11)(12,13), (4,6)(7,8)(9,13)(10,11)(12,14), (7,8)(9,12)(10,11)(13,14));