Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,7,10,6,11,5,19)(3,12,21,23,15,28,14,8)(4,9,20,24,18,25,17,26)(13,27,30,22,16,31,29,32)(33,34,35);(1,3,5,14,6,15,7,21)(2,8,10,12,11,23,19,28)(4,16,17,30,18,13,20,29)(9,22,24,31,25,32,26,27);(1,4,6,18)(2,9,11,25)(3,13,15,16)(5,20,7,17)(8,22,23,32)(10,26,19,24)(12,27,28,31)(14,30,21,29)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5361);