Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#1867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #120 |
| commutator | C23 | C22×C32×C4 |
| Frattini, cosocle | C24 | C23×C32 |
| prime radical | C22×C32×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C24×C32 ◁ G | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(7,8,10,9)(15,18);(4,5,6)(7,9,10,8)(11,12)(13,15)(14,16)(17,18);(11,13)(12,15)(14,17)(16,18)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2927);