Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| prime radical | C32×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(8,13)(9,16)(10,11)(15,18)(17,19)(20,21)(24,28)(26,33)(29,37)(30,34)(31,35)(36,38);(4,5,6)(7,8)(9,14)(10,18)(11,13)(12,15)(16,21)(17,22)(19,20)(23,24,27,30)(25,29,32,38)(26,34,33,28)(31,36,35,37);(7,9,12,17)(8,14,15,22)(10,16,11,19)(13,20,18,21)(23,25,27,32)(24,29,30,38)(26,35,33,31)(28,36,34,37)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2718);