Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1304 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #95 |
| commutator | C23 | C42×C9 |
| Frattini, prime radical, cosocle | C25×C3 | C22×C3 |
| socle | C22×C3 | C3×#95 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C25×C9 ◁ G | |
594 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,18,20,23)(13,16)(14,17)(15,19,22,25)(21,24);(10,12)(11,15,17,23)(13,20)(14,19)(16,22,24,18)(21,25)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 36);