Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of C7⋊(C7⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2×C7 |
| unambiguous semidirect product | C7⋊(C7⋊S4) |
| unambiguous semidirect product | A4⋊(C7⋊D7) |
| Fitting, socle | C22×C72 ⋊ D3 |
| derived | C72×A4 ⋊ C2 |
| Sylow | C72 ⋊ S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C72×A4 | C2 |
| Fitting, socle | C22×C72 | D3 |
| LCS | G ▷ C72×A4 | |
| UCS | C1 | |
110 conjugacy classes of subgroups.
| transitive degree | μt(G) = 196 |
Polycyclic presentation:
2,3,7^2,2^2|b^a=b^2,c^a=c^6,d^a=d^6,e^a=ef,e^b=f,f^b=ef101 conjugacy classes — view the full character table.
| maximal subgroup of | #72214642, #72214719, S4×C7⋊D7, … |
| maximal normal subgroup of | #72214719, S4×C7⋊D7, #72222205, … |
| maximal quotient of | #72214642 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 242);