Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#33855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C7 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#443 ◁ G | |
640 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)(8,9,15,21)(10,22,27,55)(11,28,31,51)(12,32,37,49)(13,20,14,19)(16,40,44,25)(17,38,46,30)(18,35,50,36)(23,56,60,42)(24,61,64,41)(26,65,39,47)(29,67,62,45)(33,52,68,43)(34,48,69,71)(53,59,70,58)(54,57,66,63);(8,10,15,27)(9,16,21,44)(11,23,31,60)(12,33,37,68)(13,26,14,39)(17,41,46,61)(18,47,50,65)(19,43,20,52)(22,49,55,32)(24,62,64,29)(25,35,40,36)(28,66,51,54)(30,58,38,59)(34,57,69,63)(42,67,56,45)(48,53,71,70);(8,11)(9,17)(10,23)(12,29)(13,38)(14,30)(15,31)(16,41)(18,45)(19,51)(20,28)(21,46)(22,53)(24,33)(25,57)(26,59)(27,60)(32,48)(34,36)(35,69)(37,62)(39,58)(40,63)(42,47)(43,54)(44,61)(49,71)(50,67)(52,66)(55,70)(56,65)(64,68);(8,11)(9,17)(10,24)(12,34)(13,30)(14,38)(15,31)(16,42)(18,48)(19,28)(20,51)(21,46)(22,54)(23,33)(25,59)(26,57)(27,64)(29,36)(32,45)(35,62)(37,69)(39,63)(40,58)(41,47)(43,53)(44,56)(49,67)(50,71)(52,70)(55,66)(60,68)(61,65)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26843);