Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#21100 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C7×C8 ◁ G | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,21,18,22,16,32)(14,23,27,34,25,37,26,19)(15,28,31,35,29,42,30,20)(24,38,40,43,41,33,39,36);(12,14)(13,19,22,34)(15,24)(16,26)(17,27)(18,25)(20,33,35,38)(21,23,32,37)(28,36,42,43)(29,41)(30,39)(31,40);(9,11)(12,15,16,30,18,29,17,31)(13,20,21,28,22,35,32,42)(14,24,26,39,25,41,27,40)(19,33,23,36,34,38,37,43);(14,25)(15,29)(19,34)(20,35)(23,37)(26,27)(28,42)(30,31)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14088);