Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| socle | C2×C32 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G | |
1356 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)(9,19)(10,24)(11,16)(12,17)(13,18)(14,21)(15,25)(20,34)(22,30)(23,29)(26,31)(27,32)(28,36)(33,35)(37,38);(4,5,6)(7,9,11,21,13,23,12,22)(8,14,17,19,18,30,16,29)(10,20,25,35,26,36,27,37)(15,28,24,33,32,34,31,38);(7,10)(8,15)(9,20)(11,25)(12,27)(13,26)(14,28)(16,31)(17,24)(18,32)(19,33)(21,35)(22,37)(23,36)(29,38)(30,34);(7,11,13,12)(8,16,18,17)(9,21,23,22)(10,25,26,27)(14,29,30,19)(15,31,32,24)(20,35,36,37)(28,38,34,33)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4127);