Finite Group DatabasebetaFinite Group Databasebeta| order | 294 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊D7 |
| automorphism | Aut(G) = AGL(1,𝔽7)2 (order 1764) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D7 |
| Fitting, socle, derived, Sylow | C72 ⋊ C2×C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2 |
| Hall | C7⋊D7 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,5)(6,7)(9,12,13,10,11,14);(1,2,4,6,7,5,3)(8,9,11,14,13,12,10)Polycyclic presentation:
2,3,7^2|c^a=c^6,d^a=d^6,d^b=d^222 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(294, 10);