Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2033 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
382 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)(8,9,14,19)(10,20,25,40)(11,26,30,61)(12,32,34,17)(13,35,37,66)(15,24,41,22)(16,42,45,29)(18,47,49,33)(21,52,27,63)(23,39,58,70)(28,62,31,48)(36,46,65,44)(38,56,43,68)(50,64,71,54)(51,55,67,57)(53,69,59,60);(8,10)(9,15)(11,27)(12,24)(13,23)(14,25)(16,43)(17,40)(18,39)(19,41)(20,32)(21,30)(22,34)(26,60)(28,53)(29,64)(31,59)(33,57)(35,67)(36,68)(37,58)(38,45)(42,54)(44,50)(46,71)(47,55)(48,52)(49,70)(51,66)(56,65)(61,69)(62,63);(8,11,12,28,14,30,34,31)(9,16,17,44,19,45,32,46)(10,21,22,53,25,27,24,59)(13,36,33,42,37,65,47,29)(15,38,20,50,41,43,40,71)(18,48,35,61,49,62,66,26)(23,56,55,54,58,68,57,64)(39,63,51,69,70,52,67,60)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6378);