Finite Group DatabasebetaFinite Group Databasebeta| order | 456 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×D4×C9 (order 864) · Inn(G) = C2×D3 · Out(G) = C23×C9 |
| homology | Abelianization (H1) = C22×C19 · Schur multiplier (H2) = C2 |
| direct product | C19×D12 |
| unambiguous semidirect product | C4⋊(D3×C19) |
| Fitting | C3×C4×C19 ⋊ C2 |
| Hall | C3×C19 ⋊ D4 |
| Sylow | C3 ⋊ D4×C19 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C19 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C19 |
| Frattini | C2 | C2×D3×C19 |
| Fitting | C3×C4×C19 | C2 |
| socle | C2×C3×C19 | C22 |
| hypercenter | C4×C19 | D3 |
| hypocenter | C3 | D4×C19 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C19 ◁ C4×C19 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 228 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(4,7,6,5)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(2,3)(5,7)171 conjugacy classes — view the full character table.
| maximal subgroup of | #11744818, #11744820, #11744822, … |
| maximal normal subgroup of | #11744818, #11744820, #11744822, … |
| maximal quotient of | C19×#204, C19×D24, C19×Dic12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(456, 31);