Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13397 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2034 |
| 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 | |
390 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)(8,9,15,21)(10,16,25,41)(11,26,28,60)(12,32,36,48)(13,20,14,19)(17,42,44,29)(18,34,49,35)(22,51,52,71)(23,54,57,70)(24,50,37,40)(27,62,58,59)(30,61,31,47)(33,46,64,45)(38,66,67,53)(39,65,68,56)(43,63,69,55);(8,10)(9,16)(11,27)(12,23)(13,37)(14,24)(15,25)(17,43)(18,39)(19,50)(20,40)(21,41)(22,30)(26,59)(28,58)(29,63)(31,52)(32,54)(33,53)(34,65)(35,56)(36,57)(38,45)(42,55)(44,69)(46,67)(47,51)(48,70)(49,68)(60,62)(61,71)(64,66);(8,11,14,30,15,28,13,31)(9,17,20,45,21,44,19,46)(10,22,24,58,25,52,37,27)(12,33,35,42,36,64,34,29)(16,38,40,69,41,67,50,43)(18,47,32,60,49,61,48,26)(23,55,56,66,57,63,65,53)(39,62,54,71,68,59,70,51)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6385);