Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1879 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
456 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)(8,9,13,19,38,41,36,18,15,21,35,52,14,20,12,34)(10,22,25,16,39,43,24,53,27,56,63,45,26,50,37,44)(11,28,31,49,40,51,29,48,33,46,66,17,32,47,30,54)(23,42,59,55,64,71,58,67,61,70,62,65,60,69,57,68);(8,10,14,39,15,27,38,26)(9,16,41,44,21,45,20,53)(11,23,40,60,33,61,32,64)(12,25,35,24,36,63,13,37)(17,42,48,71,49,70,54,69)(18,50,52,56,34,43,19,22)(28,65,47,68,46,55,51,67)(29,62,66,58,30,59,31,57);(8,11)(9,17)(10,23)(12,31)(13,30)(14,40)(15,33)(16,42)(18,51)(19,46)(20,54)(21,49)(22,55)(24,62)(25,57)(26,64)(27,61)(28,52)(29,35)(32,38)(34,47)(36,66)(37,59)(39,60)(41,48)(43,68)(44,71)(45,70)(50,67)(53,69)(56,65)(58,63)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6019);