Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C32×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C8 | D4 |
| commutator | C4 | C22×C32×C8 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C22×C32 | C4⋊C8 |
| prime radical | C23×C32 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C8 ◁ C22×C32×C8 ◁ G | |
504 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);(3,4,7,12,8,11,16,18,9,13,15,10,6,17,5,14)(22,23,24);(3,5,6,15,9,16,8,7)(4,10,11,14,13,12,17,18)504 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C3⋊D3×#364, C2×#11772190 |
| maximal subgroup of | C2×C32×#1369, C2×C3⋊D3×#364, C2×#11772190 |
| maximal normal subgroup of | C2×C32×#1369, C2×C3⋊D3×#364, C2×#11772190 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3212);