Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C7 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C4×C7×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7×C8 | C22 |
| commutator | C2 | C23×C42×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7×C8 ◁ G | |
3366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(23,24,25,26,27,28,29);(3,4,5,6);(7,8,10,13,11,14,17,20)(9,12,16,19,15,18,21,22);(7,9)(8,12)(10,16)(11,15)(13,19)(14,18)(17,21)(20,22);(9,15)(12,18)(16,21)(19,22)1120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53043);