Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C2×C9×#475 |
| 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 | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(19,20,21,22,23,24,25,26,27);(3,4,6,8)(5,9,10,7)(12,15)(13,17)(16,18);(3,5,6,10)(4,7,8,9);(11,12,14,16)(13,18,17,15)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1805);