Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13035 |
| 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)(8,9,13,20,38,41,37,18,15,22,36,52,14,21,12,19)(10,23,27,16,39,43,25,53,29,56,63,45,28,50,26,44)(11,30,32,49,40,51,31,48,34,46,35,17,33,47,65,54)(24,42,59,67,64,70,58,66,61,69,62,55,60,68,57,71);(8,10,14,39,15,29,38,28)(9,16,41,44,22,45,21,53)(11,24,40,60,34,61,33,64)(12,27,36,25,37,63,13,26)(17,42,48,70,49,69,54,68)(18,50,52,56,19,43,20,23)(30,55,47,71,46,67,51,66)(31,62,35,58,65,59,32,57);(8,11)(9,17)(10,24)(12,35)(13,31)(14,40)(15,34)(16,42)(18,47)(19,51)(20,30)(21,54)(22,49)(23,55)(25,59)(26,62)(27,58)(28,64)(29,61)(32,37)(33,38)(36,65)(39,60)(41,48)(43,66)(44,70)(45,69)(46,52)(50,71)(53,68)(56,67)(57,63)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6023);