Finite Group DatabasebetaFinite Group Databasebeta| order | 8708 |
| automorphism | Aut(G) order 8098440 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic2177 |
| unambiguous semidirect product | C7⋊Dic311 |
| unambiguous semidirect product | C311⋊Dic7 |
| derived, Hall | C7×C311 ⋊ C4 |
| Sylow | C311 ⋊ Dic7 |
| Sylow | C7 ⋊ Dic311 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D2177 |
| commutator, hypocenter | C7×C311 | C4 |
| Fitting, socle | C2×C7×C311 | C2 |
| LCS | G ▷ C7×C311 | |
| UCS | C1 ◁ C2 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
2,7,311,2|a^2=d,b^a=b^6,c^a=c^3102180 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×Dic2177, Dic6531, ΓL(1,𝔽8)⋊Dic311, … |
| maximal normal subgroup of | C3×Dic2177, ΓL(1,𝔽8)⋊Dic311, Dic4354 |
| maximal quotient of | C3×Dic2177, Dic6531, Dic15239 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8708, 4);