Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12832 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #427 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1807 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×M16 ◁ C2×C7×#131 ◁ G | |
358 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)(10,20,23,41)(11,25,27,16)(12,29)(13,32)(14,18)(15,22,35,47)(17,30)(19,33)(21,43)(24,50,49,42)(26,54,55,58)(28,38,57,53)(31,59)(34,44)(36,63,61,48)(37,56,52,70)(39,60)(40,46)(45,64)(51,68)(62,71)(65,67)(66,69);(8,10,12,22,13,24,30,48,14,23,31,47,33,49,60,63)(9,15,17,20,19,36,29,42,18,35,39,41,32,61,59,50)(11,21,26,44,28,45,56,67,27,46,55,68,57,69,70,71)(16,34,37,40,38,62,58,64,25,51,52,43,53,65,54,66);(8,11)(9,16)(10,21)(12,26)(13,28)(14,27)(15,34)(17,37)(18,25)(19,38)(20,40)(22,44)(23,46)(24,45)(29,58)(30,56)(31,55)(32,53)(33,57)(35,51)(36,62)(39,52)(41,43)(42,64)(47,68)(48,67)(49,69)(50,66)(54,59)(60,70)(61,65)(63,71)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5820);