Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1840 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C23 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle, prime radical | C23×C32 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ G | |
588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(7,8)(9,14)(10,12)(11,13)(15,16)(17,19)(18,20)(21,22)(23,24,26,29,27,30,33,36)(25,28,32,35,31,34,37,38);(4,5,6)(7,9,11,10)(8,12,13,14)(15,17,18,21)(16,19,20,22)(23,25)(24,28)(26,32)(27,31)(29,35)(30,34)(33,37)(36,38);(7,10,11,9)(8,12,13,14)(16,20)(19,22)(25,31)(28,34)(32,37)(35,38)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2900);