Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C22×C3×C42 |
| direct product | C32×#2500 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| prime radical | C2×C32×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(8,10)(11,12)(13,17)(14,15)(16,18)(19,20,22,24)(21,25,26,23);(4,5,6)(12,15)(16,17)(19,20)(21,23)(22,24)(25,26);(7,8,9,10)(11,12)(13,16)(14,15)(17,18);(11,13,14,18)(12,16,15,17)(19,21,22,26)(20,23,24,25)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3560);