Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C22×C3×C4 |
| direct product | C32×#1393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #99 |
| commutator | C2×C4 | C32×C42 |
| Frattini, cosocle | C22×D4 | C22×C32 |
| prime radical | C22×C32×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C23×C32 ◁ C22×C32×D4 ◁ G | |
696 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3)(11,12)(13,16,18,15)(14,17);(4,5,6)(7,8,9,10)(12,15,17,16)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2453);