Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#3284 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C4×D4 ◁ G | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(7,8)(9,10)(11,12,17,14)(13,21,24,26)(15,20,16,19)(18,25,22,23);(4,5,6)(11,13,17,24)(12,18,14,22)(15,23,16,25)(19,21,20,26);(8,10)(11,14,17,12)(13,22,24,18)(15,20,16,19)(21,23,26,25);(11,15)(12,19)(14,20)(16,17)(21,26)(23,25)315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4344);