Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#21780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2308 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3486 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,16,27,31)(10,18,21,32,20,14,25,29)(11,22,23,33,24,15,26,30)(19,28,36,38,35,34,37,39);(8,10)(9,14)(11,19)(12,25)(13,20)(15,28)(16,18)(17,32)(21,27)(22,34)(23,37)(24,35)(26,36)(29,31)(30,39)(33,38);(14,18)(28,34)(29,32)(38,39);(8,11,13,24)(9,15,16,22)(10,19,20,35)(12,26,27,23)(14,28,18,34)(17,33,31,30)(21,37,25,36)(29,39,32,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14768);