Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#33857 |
| 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 | |
646 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)(10,22,27,57)(11,28,32,53)(12,33,38,51)(13,20,14,19)(16,41,45,25)(17,39,48,31)(18,36,52,37)(23,43,29,66)(24,42,63,46)(26,64,40,49)(30,67,68,47)(34,54,58,44)(35,50,69,71)(55,61,70,60)(56,62,65,59);(8,10)(9,16)(11,29)(12,34)(13,26)(14,40)(15,27)(17,46)(18,49)(19,44)(20,54)(21,45)(22,33)(23,32)(24,30)(25,37)(28,65)(31,61)(35,62)(36,41)(38,58)(39,60)(42,48)(43,47)(50,70)(51,57)(52,64)(53,56)(55,71)(59,69)(63,68)(66,67);(8,11)(9,17)(10,23)(12,30)(13,39)(14,31)(15,32)(16,42)(18,47)(19,53)(20,28)(21,48)(22,55)(24,58)(25,59)(26,61)(27,29)(33,50)(34,63)(35,37)(36,69)(38,68)(40,60)(41,62)(43,64)(44,65)(45,46)(49,66)(51,71)(52,67)(54,56)(57,70);(8,11)(9,17)(10,24)(12,35)(13,31)(14,39)(15,32)(16,43)(18,50)(19,28)(20,53)(21,48)(22,56)(23,58)(25,60)(26,62)(27,63)(29,34)(30,37)(33,47)(36,68)(38,69)(40,59)(41,61)(42,64)(44,70)(45,66)(46,49)(51,67)(52,71)(54,55)(57,65)301 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26845);