Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C9×#3011 |
| 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 | C2×C3×2−1+4 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C9×#41 ◁ G | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,19,23)(16,24,27,37)(17,21,28,35)(18,30,32,22)(20,31,34,26)(25,38,39,45)(29,42,41,36)(33,43,44,40);(14,16,19,27)(15,20,23,34)(17,25,28,39)(18,31,32,26)(21,33,35,44)(22,37,30,24)(29,43,41,40)(36,45,42,38);(11,13)(14,17)(15,21)(16,25)(18,29)(19,28)(20,33)(22,36)(23,35)(24,38)(26,40)(27,39)(30,42)(31,43)(32,41)(34,44)(37,45);(17,28)(21,35)(25,39)(29,41)(33,44)(36,42)(38,45)(40,43)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1743);