Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#60472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×#41 ◁ G | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,15,21)(10,22,27,38)(11,28,31,45)(12,18,32,42)(13,20,14,19)(16,26,39,25)(17,35,41,30)(23,48,51,36)(24,47,52,62)(29,56,57,65)(33,44,34,43)(37,54,61,53)(40,59,64,58)(46,55,63,50)(49,67,68,60)(66,70,71,69);(8,10,14,26,15,27,13,25)(9,16,20,38,21,39,19,22)(11,23,35,55,31,51,30,50)(12,24,34,54,32,52,33,53)(17,36,45,63,41,48,28,46)(18,37,44,62,42,61,43,47)(29,49,59,70,57,68,58,69)(40,60,65,71,64,67,56,66);(8,11)(9,17)(10,23)(12,29)(13,30)(14,35)(15,31)(16,36)(18,40)(19,28)(20,45)(21,41)(22,46)(24,49)(25,50)(26,55)(27,51)(32,57)(33,58)(34,59)(37,60)(38,63)(39,48)(42,64)(43,56)(44,65)(47,66)(52,68)(53,69)(54,70)(61,67)(62,71);(8,12)(9,18)(10,24)(11,29)(13,33)(14,34)(15,32)(16,37)(17,40)(19,43)(20,44)(21,42)(22,47)(23,49)(25,53)(26,54)(27,52)(28,56)(30,58)(31,57)(35,59)(36,60)(38,62)(39,61)(41,64)(45,65)(46,66)(48,67)(50,69)(51,68)(55,70)(63,71);(12,32)(18,42)(24,52)(29,57)(33,34)(37,61)(40,64)(43,44)(47,62)(49,68)(53,54)(56,65)(58,59)(60,67)(66,71)(69,70)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53460);