Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C7×S4 (order 1008) · Inn(G) = C22 · Out(G) = C2×C3×D3×C7 |
| homology | Abelianization (H1) = C22×C49 · Schur multiplier (H2) = C1 |
| direct product | Q8×C49 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C49 | C22 |
| commutator | C2 | C22×C49 |
| Frattini, socle, prime radical, cosocle | C2×C7 | C22×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C49 ◁ G | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 2 generators on 57 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(1,5,3,7)(2,6,4,8)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 10);