Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×C7×D4 (order 336) · Inn(G) = C1 · Out(G) = C2×C3×C7×D4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C4×C49 |
| multiplicative | (ℤ/591)ˣ, (ℤ/788)ˣ, (ℤ/1182)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C7 | C22×C7 |
| socle, prime radical | C22×C7 | C2×C7 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 2 generators on 55 points:
(1,2,3,4)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(5,6)392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 8);