Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2021 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #120 |
| commutator | C24 | C23×C9 |
| Frattini, cosocle | C24×C3 | C23×C3 |
| socle | C22×C3 | C3×#120 |
| prime radical | C3×#2021 | C3 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C24×C9 ◁ G | |
957 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 957 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15)(16,18)(17,19)(20,21);(18,21);(11,13)(14,16)(15,18)(17,20)(19,21)234 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268822, D9×#2021 |
| maximal subgroup of | #72268822, D9×#2021, C9×#23900 |
| maximal normal subgroup of | #72268822, D9×#2021, C9×#23900 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 753);