Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of C7⋊#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C7⋊#157 |
| unambiguous semidirect product | (C2×C7)⋊#157 |
| Fitting, socle | C2×C32×C7 ⋊ C4 |
| derived, Hall | C32×C7 ⋊ C2×C4 |
| Sylow | C32 ⋊ C2×Dic7 |
| Sylow | C7 ⋊ C2×#157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C7⋊#157 |
| commutator, hypocenter | C32×C7 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C7 | C4 |
| prime radical | C2×C7×C3⋊D3 | C2 |
| cosocle | C7×C3⋊D3 | C22 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 15 points:
(3,4)(5,7,8,6)(10,11)(12,13)(14,15);(1,2)(3,5,8)(4,7,6)(9,10,12,14,15,13,11)48 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×#157, (C2×D7)⋊#157, Dic7⋊#157, … |
| maximal normal subgroup of | Dic7×#157, (C2×D7)⋊#157, Dic7⋊#157, … |
| maximal quotient of | #11759521, #11759522, C4×C7⋊#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 188);