Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C22×((C3⋊D3)⋊C3)⋊C2 (order 432) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C4×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | (C4×C3⋊D3)⋊C3 |
| Fitting | C4×H(3,𝔽3) ⋊ C2 |
| derived | C32 ⋊ C2×C3×C4 |
| Sylow | H(3,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C2×C3×C4 |
| Frattini, socle | C2×C3 | C2×C3×D3 |
| Fitting | C4×H(3,𝔽3) | C2 |
| prime radical | C2×(C3⋊D3)⋊C3 | C2 |
| cosocle | C2×C32 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(6,11,13,8,9,12)(7,10);(1,2,3,4)(5,6,8)(7,9,12)(10,13,11)40 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C3)⋊C8, #88430, C8×(C3⋊D3)⋊C3, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C3)⋊C8, #88430, C8×(C3⋊D3)⋊C3, … |
| maximal quotient of | C8×(C3⋊D3)⋊C3, #3743⋊C3, C4×(C3⋊Dic3)⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 50);