Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #120 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C2×C32 | #386 |
| prime radical | C32×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8)(9,18,21,29)(10,22)(11,16)(12,26)(13,17)(14,20,30,25)(15,23)(19,34,37,28)(24,31)(27,32)(33,35,38,36);(4,5,6)(7,9,11,25,13,21,12,20)(8,14,16,29,17,30,26,18)(10,19,23,36,24,37,27,35)(15,28,31,38,32,34,22,33);(7,10)(8,15)(9,19)(11,23)(12,27)(13,24)(14,28)(16,31)(17,32)(18,33)(20,35)(21,37)(22,26)(25,36)(29,38)(30,34)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2941);