Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽11) (order 1320) · Inn(G) = D11 · Out(G) = C22×C3×C5 |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×Dic11 |
| unambiguous semidirect product | C11⋊(C4×C9) |
| derived, Sylow | C11 ⋊ C4×C9 |
| Hall | C9×C11 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D11 |
| commutator, hypocenter | C11 | C4×C9 |
| Frattini | C2×C3 | C3×D11 |
| Fitting | C2×C9×C11 | C2 |
| socle, prime radical, cosocle | C2×C3×C11 | C2×C3 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C9 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 396 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,14,15)(16,17,19,18,20,22,21,23,24);(1,2,4,6,8,10,11,9,7,5,3)126 conjugacy classes — view the full character table.
| maximal subgroup of | #11716175, D9×Dic11, D99⋊C4, … |
| maximal normal subgroup of | #11716175, D9×Dic11, D99⋊C4, … |
| maximal quotient of | C9×C11⋊C8, C2×C9×Dic11, C27×Dic11, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 2);