Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#3244 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C3×#121 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,18)(13,16)(14,17)(15,20)(19,24)(21,23)(22,25)(26,27)(28,35)(29,33)(30,32)(31,36)(34,38)(37,39)(40,41);(10,12,14,20)(11,15,17,18)(13,21,22,19)(16,24,25,23)(26,28)(27,31)(29,36)(30,37)(32,40)(33,35)(34,39)(38,41);(10,13,14,22)(11,16,17,25)(12,19,20,21)(15,23,18,24)(27,32)(28,36)(30,38)(31,39)(33,34)(35,40);(11,17)(15,18)(16,25)(23,24)(27,33)(28,36)(32,34)(37,41)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1976);