Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1296 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C22⋊C4 |
| commutator | C2×C4 | C32×C42 |
| Frattini, cosocle | C2×C42 | C22×C32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C2×C32×C42 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(7,8,9,10)(12,16)(13,14)(17,18)(19,20)(21,23)(22,25)(24,26);(4,5,6)(11,12,14,17,15,18,13,16)(20,23)(21,26)(24,25)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2356);