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×#1294 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C22⋊C4 |
| commutator | C2×C4 | C32×C42 |
| Frattini, cosocle | C23×C4 | C22×C32 |
| prime radical | C24×C32 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(7,8,9,10)(11,12)(13,15)(14,17)(16,18)(19,20)(21,25)(22,24)(23,26);(4,5,6)(12,15,17,16)(19,21)(20,23,24,25)(22,26)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2354);