Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×#360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #102 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| prime radical | C2×C32×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(19,20,21);(4,8,9,17)(5,11)(6,14)(10,12,15,18)(13,16)(22,23,24);(3,4)(5,12)(6,15)(7,9)(8,16)(10,14)(11,17)(13,18)288 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C32×AGL(1,ℤ/16), C2×#11797944, C2×C3⋊D3×#360, … |
| maximal normal subgroup of | C2×C32×AGL(1,ℤ/16), C2×#11797944, C2×C3⋊D3×#360, … |
| maximal quotient of | C2×C32×#1333 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3204);