Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2957 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | C22×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C2×C3×C4×D4 |
| prime radical | C2×C3×#135 | C2×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C2×C9×M16 ◁ G | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,15,20,16,19,14,28)(12,21,23,17,24,32,29,31)(13,18,27,33,25,34,26,38)(22,35,37,30,36,40,39,41);(10,12)(11,17)(13,22)(14,29)(15,23)(16,24)(18,30)(19,31)(20,32)(21,28)(25,36)(26,39)(27,37)(33,40)(34,41)(35,38);(10,13,15,27,16,25,14,26)(11,18,20,33,19,34,28,38)(12,22,23,37,24,36,29,39)(17,30,32,40,31,41,21,35);(11,19)(13,25)(17,31)(20,28)(21,32)(22,36)(26,27)(37,39)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1689);