Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C4 |
| direct product | C9×#1572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C3×C42 | C23×C3 |
| socle | C22×C3 | C2×C3×C4⋊C4 |
| prime radical | C23×C3 | C3×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C2×C42×C9 ◁ G | |
225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,19)(13,17)(14,23)(15,18)(16,21)(20,24)(22,25)(26,27,29,32,30,33,36,39)(28,31,35,38,34,37,40,41);(10,12,14,21,15,22,13,20)(11,16,17,19,18,24,23,25)(26,28)(27,31)(29,35)(30,34)(32,38)(33,37)(36,40)(39,41);(10,13,15,14)(11,17,18,23)(12,20,22,21)(16,19,24,25)(28,34)(31,37)(35,40)(38,41)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 304);