Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#1814 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #121 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C2×C32 | #380 |
| prime radical | C2×C32×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
768 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)(10,23,25,30)(11,26)(12,16)(13,17)(14,20)(15,24,31,27)(19,28,35,38)(21,29)(22,33)(32,34,37,36);(4,5,6)(7,9,11,20,13,22,12,21)(8,14,16,18,17,29,26,33)(10,19,24,34,25,35,27,36)(15,28,30,32,31,38,23,37);(7,10)(8,15)(9,19)(11,24)(12,27)(13,25)(14,28)(16,30)(17,31)(18,32)(20,34)(21,36)(22,35)(23,26)(29,38)(33,37)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2874);