Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C72 · Schur multiplier (H2) = C7 |
| direct product | C72×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C72 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C72 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C72 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5,6,7)(15,16,17,18)(19,22,21,20);(8,9,10,11,12,13,14)(15,19,17,21)(16,20,18,22)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 35);