Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#3044 |
| 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×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
687 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)(11,13)(14,15,19,23)(16,20,25,34)(17,27,29,37)(18,30,32,22)(21,31,36,28)(24,38,39,45)(26,42,41,35)(33,43,44,40);(10,11,12,13)(14,16)(15,20)(17,24)(18,26)(19,25)(21,33)(22,35)(23,34)(27,38)(28,40)(29,39)(30,42)(31,43)(32,41)(36,44)(37,45);(14,17,19,29)(15,21,23,36)(16,24,25,39)(18,31,32,28)(20,33,34,44)(22,37,30,27)(26,43,41,40)(35,45,42,38);(16,25)(20,34)(24,39)(26,41)(33,44)(35,42)(38,45)(40,43)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1776);