Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C2×C9×#473 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #121 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C3×#121 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C22×C4×C9 ◁ G | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2)(35,36,37,38,39,40,41,42,43);(3,4,9,14)(5,15,19,25)(6,20,22,28)(7,13,8,12)(10,18,26,17)(11,23,27,21)(16,30,32,24)(29,33,34,31);(3,5,8,18,9,19,7,17)(4,10,13,25,14,26,12,15)(6,16,23,33,22,32,21,31)(11,24,28,34,27,30,20,29);(3,6)(4,11)(5,16)(7,21)(8,23)(9,22)(10,24)(12,20)(13,28)(14,27)(15,29)(17,31)(18,33)(19,32)(25,34)(26,30)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1798);