Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| prime radical | C32×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32×C4 ◁ G | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2,3)(7,8,13,18)(9,19,21,30)(10,24,26,32)(11,17,12,16)(14,23,29,22)(15,27,31,25)(20,34,35,28)(33,37,38,36);(4,5,6)(7,9,11,22,13,21,12,23)(8,14,16,19,18,29,17,30)(10,20,25,36,26,35,27,37)(15,28,24,33,31,34,32,38);(7,10)(8,15)(9,20)(11,25)(12,27)(13,26)(14,28)(16,24)(17,32)(18,31)(19,33)(21,35)(22,36)(23,37)(29,34)(30,38);(9,21)(14,29)(19,30)(20,35)(22,23)(28,34)(33,38)(36,37)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4129);