Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12841 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #466 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1808 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ 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)(10,21)(11,28,31,42)(12,32)(13,19)(14,36)(15,20)(16,24)(17,30,41,35)(18,22)(23,50,53,69)(25,46)(26,57)(27,47)(29,40,60,67)(33,43)(34,62)(37,64,65,71)(38,54)(39,55)(44,52,68,56)(45,48)(49,59,63,61)(51,58,70,66);(8,10,22,48,14,26,34,55,15,27,33,54,13,25,12,24)(9,16,32,46,19,38,43,47,20,39,62,57,36,45,18,21)(11,23,49,64,30,52,60,70,31,53,63,71,35,56,29,51)(17,37,61,69,42,66,67,68,41,65,59,50,28,58,40,44);(8,11)(9,17)(10,23)(12,29)(13,35)(14,30)(15,31)(16,37)(18,40)(19,42)(20,41)(21,44)(22,49)(24,51)(25,56)(26,52)(27,53)(28,36)(32,61)(33,63)(34,60)(38,66)(39,65)(43,67)(45,58)(46,69)(47,68)(48,64)(50,57)(54,71)(55,70)(59,62)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5829);