Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13622 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #357 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×#357 |
| prime radical | C2×C7×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
294 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)(10,21,24,52)(11,17)(12,28)(13,33)(14,37)(15,20)(16,23,40,58)(18,30)(19,35)(22,50,38,54)(25,61,60,71)(26,62)(27,48)(29,47,59,51)(31,67)(32,49)(34,53,57,44)(36,45)(39,55,42,69)(41,68,70,63)(43,64)(46,56)(65,66);(8,10)(9,16)(11,22)(12,29)(13,34)(14,38)(15,24)(17,39)(18,44)(19,47)(20,40)(21,49)(23,56)(25,35)(26,63)(27,60)(28,52)(30,58)(31,68)(32,59)(33,41)(36,57)(37,42)(43,61)(45,70)(46,53)(48,51)(50,67)(54,62)(55,66)(64,69)(65,71);(8,11,15,14)(9,17,20,37)(10,22,24,38)(12,26,32,31)(13,27,36,35)(16,39,40,42)(18,43,46,65)(19,33,48,45)(21,50,52,54)(23,55,58,69)(25,34,60,57)(28,62,49,67)(29,63,59,68)(30,64,56,66)(41,51,70,47)(44,61,53,71)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6610);