Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13016 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1879 |
| prime radical | C2×C7×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,14,20,15,21,37,53,16,22,13,38,36,52,12,19)(10,23,27,46,28,51,25,54,29,57,26,17,41,44,39,45)(11,30,32,55,33,66,31,18,34,48,35,49,42,47,40,50)(24,58,60,65,61,43,63,56,62,69,67,71,64,70,59,68);(8,10,15,41,16,29,36,28)(9,17,21,54,22,46,52,45)(11,24,42,61,34,62,33,64)(12,26,14,25,37,27,13,39)(18,43,55,69,50,70,49,58)(19,51,20,23,53,44,38,57)(30,65,47,56,48,71,66,68)(31,60,32,63,40,67,35,59);(8,11)(9,18)(10,24)(12,35)(13,40)(14,31)(15,42)(16,34)(17,43)(19,30)(20,47)(21,55)(22,50)(23,56)(25,60)(26,59)(27,63)(28,64)(29,62)(32,37)(33,36)(38,66)(39,67)(41,61)(44,71)(45,58)(46,70)(48,53)(49,52)(51,65)(54,69)(57,68)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6004);