Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#9027 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7)(9,11)(12,13,14,17,15,18,19,16)(20,21)(22,25)(23,30,31,26)(24,27)(28,32,34,35)(29,33);(8,9,10,11)(13,16)(14,19)(17,18)(20,22,24,29)(21,25,27,33)(23,28,31,34)(26,32,30,35);(20,23,24,31)(21,26,27,30)(22,28,29,34)(25,32,33,35)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2015);