Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C72 ⋊ C2×C3 |
| Hall | C22⋊(C7⋊D7) ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ D4 |
| Sylow | C72 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×ΓL(1,𝔽8)⋊D7 |
| commutator, socle, cosocle | C2×C72 | C22×C3 |
| Fitting | C22×C72 | C2×C3 |
| hypercenter | C22 | ΓL(1,𝔽8)⋊D7 |
| hypocenter | C72 | C3×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C22 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2^2,3,2,7^2|b^a=bd,e^a=e^6,f^a=f^6,e^c=e^479 conjugacy classes — view the full character table.
| maximal subgroup of | #72222016, D4×ΓL(1,𝔽8)⋊D7, #72222205, … |
| maximal normal subgroup of | #72222016, D4×ΓL(1,𝔽8)⋊D7, C2×#11924202, … |
| maximal quotient of | C2×#11924202, #72245190, #72245191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 117);