Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #354 |
| 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 | |
78 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)(3,12,14,30,13,29,31,26)(5,10,20,27,21,8,17,25)(7,22,23,15,24,16,32,19)(33,34,35);(1,3,6,13)(2,7)(4,16,18,22)(5,19,21,15)(8,12)(9,26)(10,29)(11,24)(14,17,31,20)(23,25)(27,32)(28,30)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 549);