Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#1411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #353 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | C4⋊Q8 | C22×C3 |
| prime radical | C22×C3 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊Q8 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,9,6,11,18,28)(3,12,13,26,15,23,30,31)(5,10,20,27,21,8,17,25)(7,16,22,19,24,29,32,14)(33,34,35);(1,3,6,15)(2,7,11,24)(4,16,18,29)(5,19,21,14)(8,23,10,12)(9,26,28,31)(13,17,30,20)(22,27,32,25)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 547);