Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#1537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C42 | C23×C3 |
| prime radical | C23×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,26,29,32)(20,24)(21,25)(22,28,31,27)(23,30)(33,34,35);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,22)(20,27)(21,29)(23,26)(24,32)(25,31)(28,30);(17,20,21,30)(18,23,25,24)(19,27,29,28)(22,26,31,32)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4828);