Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 16128 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C7 |
| abelian | C2×C4×C72 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C72 |
| socle, prime radical | C22×C72 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(7,8,9,10,11,12,13);(5,6)(14,15,16,17,18,19,20)392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 33);