Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = S4×((C3⋊D3)⋊C3)⋊C2 (order 2592) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×(C3⋊D3)⋊C3 |
| Fitting | C22×H(3,𝔽3) ⋊ C2 |
| derived | C32 ⋊ C23×C3 |
| Sylow | H(3,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32 | C23×C3 |
| Frattini | C3 | C22×C3×D3 |
| Fitting | C22×H(3,𝔽3) | C2 |
| socle | C22×C3 | C3×D3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(6,11,13,8,9,12)(7,10);(3,4)(5,6,8)(7,9,12)(10,13,11);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | #88450, #3752⋊C3, C2×((C3⋊D3)⋊C3)⋊C4, … |
| maximal normal subgroup of | #88450, #3752⋊C3, C2×((C3⋊D3)⋊C3)⋊C4, … |
| maximal quotient of | C2×#634⋊C3, C2×C4×(C3⋊D3)⋊C3, C2×C4⋊((C3⋊D3)⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 110);