Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13021 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1879 |
| prime radical | C7×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
462 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)(8,9,14,21,15,22,38,53,16,23,13,20,37,52,12,19)(10,24,27,46,28,51,26,56,29,54,36,17,41,44,39,45)(11,30,33,57,34,55,31,18,35,48,32,49,42,47,40,50)(25,59,61,66,62,43,67,68,63,69,64,71,65,70,60,58);(8,10,15,41,16,29,37,28)(9,17,22,56,23,46,52,45)(11,25,42,62,35,63,34,65)(12,36,14,26,38,27,13,39)(18,43,57,69,50,70,49,59)(19,51,21,24,53,44,20,54)(30,66,47,68,48,71,55,58)(31,61,33,67,40,64,32,60);(8,11)(9,18)(10,25)(12,33)(13,31)(14,40)(15,42)(16,35)(17,43)(19,48)(20,47)(21,55)(22,57)(23,50)(24,58)(26,64)(27,60)(28,65)(29,63)(30,53)(32,38)(34,37)(36,67)(39,61)(41,62)(44,66)(45,59)(46,70)(49,52)(51,71)(54,68)(56,69)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6009);