Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×C9×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C9 | C23 |
| commutator | C22 | C25×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C24×C3 | C23×C3 |
| prime radical | C22×C3×#127 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C9 ◁ G | |
4500 conjugacy classes of subgroups; only the 1668 normal subgroups are shown.
This group has 4500 conjugacy classes of subgroups — too many to draw. Only its 1668 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(13,14,15,16,17,18,19,20,21);(3,4);(6,8)(10,11);(9,10,12,11);(5,6,7,8)(9,11,12,10)504 conjugacy classes — view the full character table.
| Fitting subgroup of | C22×D9×#127, #72268850 |
| maximal subgroup of | C22×D9×#127, #72268850, C9×#15258 |
| maximal normal subgroup of | C22×D9×#127, #72268850, C9×#15258 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2172);