Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×#504 (order 256) · Inn(G) = C22 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C3×C32 · Schur multiplier (H2) = C1 |
| direct product | C3×M128 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C32 | C22 |
| commutator | C2 | C2×C3×C32 |
| Frattini, cosocle | C32 | C22×C3 |
| socle | C2×C3 | C2×C32 |
| prime radical | C22×C3 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C32 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(5,37)(7,39)(9,41)(11,43)(13,45)(15,47)(17,49)(19,51)(21,53)(23,55)(25,57)(27,59)(29,61)(31,63)(33,65)(35,67)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 564);