Finite Group DatabasebetaFinite Group Databasebeta| order | 368 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C11×#120 (order 704) · Inn(G) = C22 · Out(G) = C2×D4×C11 |
| homology | Abelianization (H1) = C2×C4×C23 · Schur multiplier (H2) = C22 |
| direct product | C22⋊C4×C23 |
| unambiguous semidirect product | C22⋊(C4×C23) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | C22 |
| commutator | C2 | C2×C4×C23 |
| Frattini, cosocle | C22 | C22×C23 |
| prime radical | C23×C23 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ G | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 184 |
A minimal faithful permutation representation — 2 generators on 31 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);(6,8)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(368, 20);