Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3284 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×D4 ◁ G | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4)(5,6,10,9)(7,14,17,20)(8,12,18,13)(11,15,19,16)(21,22,23);(5,7,10,17)(6,11,9,19)(8,15,18,16)(12,14,13,20);(2,4)(5,8,10,18)(6,12,9,13)(7,15,17,16)(11,14,19,20);(5,9)(6,10)(7,14)(8,13)(11,16)(12,18)(15,19)(17,20)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17724);