Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of C3⋊D9 |
| automorphism | Aut(G) = C22×#72210336 (order 11664) · Inn(G) = C3⋊D9 · Out(G) = C22×#272 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C4×C3⋊D9 |
| Fitting | C3×C4×C9 ⋊ C2 |
| derived, Sylow | C3×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C2×C4 |
| Frattini | C2×C3 | C2×C3⋊D3 |
| Fitting | C3×C4×C9 | C2 |
| socle | C2×C32 | C2×D3 |
| prime radical | C2×C3⋊D9 | C2 |
| cosocle | C2×C3×C9 | C22 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3)(5,7)(6,10)(8,9)(11,12);(1,2,3)(13,14,15,16);(4,12,7,10,9,8,6,5,11)60 conjugacy classes — view the full character table.
| maximal subgroup of | (C3⋊D9)⋊C8, #88414, C8×C3⋊D9, … |
| maximal normal subgroup of | (C3⋊D9)⋊C8, #88414, C8×C3⋊D9, … |
| maximal quotient of | C8×C3⋊D9, #88523, C4×C3⋊Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 64);