Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12830 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #418 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1804 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
378 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)(8,9,13,20,15,18,14,37)(10,21,24,42,25,16,36,41)(11,26,29,58,27,59,38,48)(12,30,33,46,35,62,34,19)(17,32,43,60,45,66,44,28)(22,50,53,71,51,67,54,69)(23,55,57,47,31,63,65,40)(39,56,68,52,49,64,70,61);(8,10,14,24,15,25,13,36)(9,16,37,41,18,21,20,42)(11,22,38,53,27,51,29,54)(12,31,34,65,35,23,33,57)(17,39,44,68,45,49,43,70)(19,47,62,63,46,40,30,55)(26,50,48,71,59,67,58,69)(28,61,66,56,60,52,32,64);(8,11)(9,17)(10,22)(12,32)(13,29)(14,38)(15,27)(16,39)(18,45)(19,48)(20,43)(21,49)(23,56)(24,53)(25,51)(26,30)(28,34)(31,64)(33,60)(35,66)(36,54)(37,44)(40,69)(41,68)(42,70)(46,58)(47,71)(50,55)(52,57)(59,62)(61,65)(63,67)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5818);