Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2190 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C32 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ C2×C32×M16 ◁ G | |
678 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 678 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) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(8,12)(9,15)(10,18)(13,20)(14,21)(16,19)(17,22);(4,5,6)(7,8)(9,16)(10,19)(11,13)(12,15)(14,18)(17,21)(20,22);(9,17)(12,20)(15,22)(16,21)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3250);