Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| isoclinism | isoclinic class of D7×C3⋊D3 |
| automorphism | Aut(G) = AGL(1,𝔽7)×AGL(2,𝔽3) (order 18144) · Inn(G) = G · Out(G) = C3×S4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | D7×C3⋊D3 |
| unambiguous semidirect product | (C3×D7)⋊D3 |
| Fitting, socle, derived, Hall | C32×C7 ⋊ C22 |
| Sylow | C32 ⋊ C2×D7 |
| Sylow | C7 ⋊ C2×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C7 | C22 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,6,7,5,3)(9,10)(12,13);(2,3)(4,5)(6,7)(11,12,13);(8,9,10)30 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#157, D32×D7, C2×D7×C3⋊D3, … |
| maximal normal subgroup of | D7×#157, D32×D7, C2×D7×C3⋊D3, … |
| maximal quotient of | D7×C3⋊Dic3, C3⋊D3×Dic7, #92675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 34);