Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C11×C3⋊D3 |
| unambiguous semidirect product | (C2×C3)⋊(D3×C11) |
| Fitting, socle | C2×C32×C11 ⋊ C2 |
| derived, Sylow | C32 ⋊ C22×C11 |
| Hall | C32×C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C22×C11 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C11 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C11 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 198 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(4,5)(7,8)(9,10,11,12,13,14,15,16,17,18,19);(1,2)(3,4,5);(6,7,8)132 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×Dic11, #11716228, #11716230, … |
| maximal normal subgroup of | C3⋊D3×Dic11, #11716228, #11716230, … |
| maximal quotient of | C11×#634, C4×C11×C3⋊D3, C11×C4⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 28);