Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#1658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32)(33,34,35);(1,2,5,8)(3,7,10,16)(4,12,11,6)(9,14,13,15)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,27)(28,30)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4949);