Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
386 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)(8,9,14,19)(10,20,25,41)(11,26,30,63)(12,31,34,17)(13,35,38,62)(15,24,42,22)(16,43,46,29)(18,48,51,33)(21,54,27,64)(23,40,36,49)(28,66,32,50)(37,47,67,45)(39,68,44,59)(52,65,69,56)(53,58,70,60)(55,71,57,61);(8,10)(9,15)(11,27)(12,24)(13,36)(14,25)(16,44)(17,41)(18,49)(19,42)(20,31)(21,30)(22,34)(23,38)(26,61)(28,55)(29,56)(32,57)(33,58)(35,53)(37,59)(39,46)(40,51)(43,65)(45,69)(47,52)(48,60)(50,54)(62,70)(63,71)(64,66)(67,68);(8,11,14,30)(9,16,19,46)(10,21)(12,32,34,28)(13,37,38,67)(15,39)(17,47,31,45)(18,50,51,66)(20,52)(22,57)(23,59)(24,55)(25,27)(26,62,63,35)(29,48,43,33)(36,68)(40,54)(41,69)(42,44)(49,64)(53,71)(56,60)(58,65)(61,70)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6393);