Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C3×C4 |
| direct product | C32×#2254 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C32 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(7,8,10,12)(9,13,14,11)(16,20)(17,24)(18,25)(21,26)(22,30)(23,29)(27,28);(4,5,6)(7,9,10,14)(8,11,12,13);(15,16,21,30,18,22,17,20,19,23,24,28,25,27,26,29)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3314);