Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C4 ◁ G | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2,3)(7,8)(9,18)(10,15)(11,28)(12,17)(13,21)(14,26)(16,29)(19,31)(20,37)(22,33)(23,32)(24,30)(25,35)(27,34)(36,38);(4,5,6)(7,9)(8,13)(10,24)(11,22)(12,23)(14,31)(15,36)(16,32)(17,33)(18,28)(19,27)(20,25)(21,29)(26,38)(30,35)(34,37);(7,10)(8,14)(9,19)(11,25)(12,27)(13,30)(15,28)(16,34)(17,35)(18,37)(20,21)(22,38)(23,24)(26,29)(31,33)(32,36);(7,10)(8,15)(9,20)(11,26)(12,27)(13,31)(14,28)(16,35)(17,34)(18,37)(19,21)(22,24)(23,38)(25,29)(30,33)(32,36)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4350);