Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C27×C3 |
| direct product | C32×C4×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C23 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C23 | C24×C32 |
| socle | C23×C32 | C24 |
| prime radical | C2×C32×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ G | |
3672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2,3)(7,8,9,10);(4,5,6)(11,12)(13,15)(14,16)(17,18);(12,15)(16,18);(12,16)(15,18)504 conjugacy classes — view the full character table.
| Fitting subgroup of | C4×C3⋊D3×#120, C3×C4×#5032 |
| maximal subgroup of | C32×AGL(1,𝔽5)×#120, C4×C3⋊D3×#120, C3×C4×#5032 |
| maximal normal subgroup of | C4×C3⋊D3×#120, C3×C4×#5032 |
| maximal quotient of | C32×AGL(1,𝔽5)×#120 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3359);