Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of AGL(1,𝔽8) |
| automorphism | Aut(G) = C7×AΓL(1,𝔽8) (order 1176) · Inn(G) = AGL(1,𝔽8) · Out(G) = C3×C7 |
| homology | Abelianization (H1) = C49 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C23⋊C49 |
| derived, Sylow | C23 ⋊ C49 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C7 | AGL(1,𝔽8) |
| commutator, hypocenter | C23 | C49 |
| Fitting, socle, prime radical, cosocle | C23×C7 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C7 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 98 |
A minimal faithful permutation representation — 2 generators on 57 points:
(2,3,4,6,8,5,7)(9,10,12,15,19,24,30,11,13,16,20,25,31,37,14,17,21,26,32,38,43,18,22,27,33,39,44,48,23,28,34,40,45,49,52,29,35,41,46,50,53,55,36,42,47,51,54,56,57);(1,5)(2,3)(4,8)(6,7)Polycyclic presentation:
7^2,2^3|a^7=b,c^a=d,d^a=e,e^a=ce56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 11);