Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | Aut(G) = C22×C3×AGL(2,𝔽3) (order 5184) · Inn(G) = C3⋊D3 · Out(G) = C22×C3×S4 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C7×C3⋊D3 |
| unambiguous semidirect product | (C2×C3)⋊(D3×C7) |
| Fitting, socle | C2×C32×C7 ⋊ C2 |
| derived, Sylow | C32 ⋊ C22×C7 |
| Hall | C32×C7 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C7 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(4,5)(7,8)(9,10,11,12,13,14,15);(1,2)(3,4,5);(6,7,8)84 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×Dic7, #92676, #92678, … |
| maximal normal subgroup of | C3⋊D3×Dic7, #92676, #92678, … |
| maximal quotient of | C7×#634, C4×C7×C3⋊D3, C7×C4⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 44);