Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C7×D4 (order 336) · Inn(G) = C22 · Out(G) = C22×C3×C7 |
| homology | Abelianization (H1) = C22×C49 · Schur multiplier (H2) = C2 |
| direct product | D4×C49 |
| unambiguous semidirect product | C4⋊(C2×C49) |
| unambiguous semidirect product | C22⋊(C2×C49) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C49 | C22 |
| commutator | C2 | C22×C49 |
| Frattini, socle, cosocle | C2×C7 | C22×C7 |
| prime radical | C7×D4 | C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C49 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 196 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(2,4)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 9);