Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C9×#491 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C2×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C2×C3×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ C2×C42×C9 ◁ G | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2)(27,28,29,30,31,32,33,34,35);(3,4)(5,8)(6,7)(9,10)(11,12,17,22)(13,19,24,25)(14,18,23,26)(15,21,16,20);(4,7)(8,10)(11,13,15,14,17,24,16,23)(12,18,20,19,22,26,21,25);(3,5,6,9)(4,8,7,10)(11,14,15,24,17,23,16,13)(12,19,20,26,22,25,21,18)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1866);