Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9×C16 · Schur multiplier (H2) = C22 |
| direct product | C9×#2258 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C32 | C22 |
| commutator | C2 | C22×C9×C16 |
| Frattini, cosocle | C3×C16 | C23×C3 |
| socle | C2×C3 | C22×C3×C16 |
| prime radical | C3×#41 | C3×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C9×C32 ◁ G | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,18,14,19,26,36,15,20,27,37,29,39,48,58,16,21,28,38,30,40,49,59,31,41,50,60,51,61,66,71)(12,17,23,33,24,34,45,55,25,35,46,56,47,57,65,70,22,32,42,52,43,53,62,67,44,54,63,68,64,69,72,73);(10,12)(11,17)(13,23)(14,24)(15,25)(16,22)(18,33)(19,34)(20,35)(21,32)(26,45)(27,46)(28,42)(29,47)(30,43)(31,44)(36,55)(37,56)(38,52)(39,57)(40,53)(41,54)(48,65)(49,62)(50,63)(51,64)(58,70)(59,67)(60,68)(61,69)(66,72)(71,73);(12,22)(17,32)(23,42)(24,43)(25,44)(33,52)(34,53)(35,54)(45,62)(46,63)(47,64)(55,67)(56,68)(57,69)(65,72)(70,73)720 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 990);