Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#22460 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
1286 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)(8,9,14,19,15,20,13,33)(10,21,24,16,25,40,34,39)(11,26,28,42,29,17,36,43)(12,18,31,47,32,46,35,45)(22,50,53,37,54,48,52,57)(23,49,55,38,56,44,30,63)(27,58,61,67,62,41,60,66)(51,65,59,64,69,68,70,71);(8,10)(9,16)(11,22)(12,30)(13,34)(14,24)(15,25)(17,37)(18,44)(19,40)(20,39)(21,33)(23,31)(26,57)(27,59)(28,53)(29,54)(32,55)(35,56)(36,52)(38,45)(41,65)(42,50)(43,48)(46,49)(47,63)(51,60)(58,68)(61,69)(62,70)(64,66)(67,71);(8,11,15,29)(9,17,20,26)(10,22,25,54)(12,27,32,62)(13,28,14,36)(16,37,39,57)(18,41,46,58)(19,42,33,43)(21,48,40,50)(23,51,56,69)(24,52,34,53)(30,59,55,70)(31,60,35,61)(38,64,63,71)(44,65,49,68)(45,66,47,67);(8,12)(9,18)(10,23)(11,27)(13,35)(14,31)(15,32)(16,38)(17,41)(19,47)(20,46)(21,49)(22,51)(24,55)(25,56)(26,58)(28,61)(29,62)(30,34)(33,45)(36,60)(37,64)(39,63)(40,44)(42,67)(43,66)(48,68)(50,65)(52,70)(53,59)(54,69)(57,71)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15448);