Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C7 |
| unambiguous semidirect product | C4⋊(C3⋊(C7⋊D7)) |
| Fitting | C3×C4×C72 ⋊ C2 |
| Hall | C3×C72 ⋊ D4 |
| Sylow | C72 ⋊ D12 |
| Sylow | C3 ⋊ C4⋊(C7⋊D7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C7⋊D7) |
| commutator, socle, cosocle | C2×C3×C72 | C22 |
| Fitting | C3×C4×C72 | C2 |
| hypercenter | C4 | C3⋊(C7⋊D7) |
| hypocenter | C3×C72 | D4 |
| LCS | G ▷ C2×C3×C72 ▷ C3×C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
160 conjugacy classes of subgroups.
| transitive degree | μt(G) = 588 |
Polycyclic presentation:
2^3,3,7^2|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^6297 conjugacy classes — view the full character table.
| maximal subgroup of | D12×C7⋊D7, D4×C3⋊(C7⋊D7), D3×C4⋊(C7⋊D7), … |
| maximal normal subgroup of | D12×C7⋊D7, D4×C3⋊(C7⋊D7), D3×C4⋊(C7⋊D7), … |
| maximal quotient of | C2×C4⋊(C3⋊(C7⋊D7)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 206);