Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#33414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22×D8 |
| commutator | C8 | C23×C4×C7 |
| Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C2×C7 | C22×#107 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×#41 ◁ C2×C7×#131 ◁ G | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,13,19,15,20,14,35)(10,21,25,46,27,47,26,55)(11,17,30,42,28,40,31,59)(12,32,33,43,34,60,22,18)(16,36,38,54,39,48,45,24)(23,44,52,67,50,65,53,70)(29,57,58,64,56,71,49,41)(37,61,63,69,62,68,66,51);(8,10,22,48,14,26,34,54,15,27,33,36,13,25,12,24)(9,16,32,46,19,38,43,47,20,39,60,55,35,45,18,21)(11,23,49,68,31,53,56,69,28,50,58,61,30,52,29,51)(17,37,57,67,42,63,64,65,40,62,71,70,59,66,41,44);(8,11)(9,17)(10,23)(12,29)(13,30)(14,31)(15,28)(16,37)(18,41)(19,42)(20,40)(21,44)(22,49)(24,51)(25,52)(26,53)(27,50)(32,57)(33,58)(34,56)(35,59)(36,61)(38,63)(39,62)(43,64)(45,66)(46,67)(47,65)(48,68)(54,69)(55,70)(60,71);(11,28)(17,40)(23,50)(29,56)(30,31)(37,62)(41,64)(42,59)(44,65)(49,58)(51,69)(52,53)(57,71)(61,68)(63,66)(67,70)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26402);