Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | Aut(G) order 2426112 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C3⋊(C3⋊Dic3) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊Dic3) |
| derived, Sylow | C33 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C3⋊D3) |
| commutator, hypocenter | C33 | C2×C4 |
| Frattini | C2 | C2×C3⋊(C3⋊D3) |
| Fitting, socle, prime radical | C22×C33 | C2 |
| cosocle | C2×C33 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(4,5)(7,8)(10,11)(12,13,14,15);(1,2)(3,4,5);(6,7,8);(9,10,11)60 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C3⋊Dic3, (C2×D3)⋊(C3⋊Dic3), #88803, … |
| maximal normal subgroup of | Dic3×C3⋊Dic3, (C2×D3)⋊(C3⋊Dic3), #88803, … |
| maximal quotient of | C2×C3⋊(C3⋊(C3⋊C8)), #88855, C4×C3⋊(C3⋊Dic3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 148);