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) = C29×C3 |
| direct product | C2×C32×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C32 | C23 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C23 | C24×C32 |
| prime radical | C26×C32 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C32 ◁ G | |
10392 conjugacy classes of subgroups; only the 1800 normal subgroups are shown.
This group has 10392 conjugacy classes of subgroups — too many to draw. Only its 1800 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(15,16,17);(4,6)(7,8,9,10)(12,14)(18,19,20);(11,12)(13,14);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3337);