Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#2398 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| socle | C23×C32 | C24 |
| prime radical | C22×C32×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ 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)(11,12,15,17)(13,18,14,16)(19,20,22,24)(21,25,26,23);(4,5,6)(7,8)(9,10)(11,13,15,14)(12,16,17,18)(19,21,22,26)(20,23,24,25);(8,10)(20,24)(23,25);(11,14,15,13)(12,16,17,18)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3458);