Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13638 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #367 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×#367 |
| prime radical | C23×C7 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
182 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)(10,21,25,48)(11,16)(12,30)(13,34)(14,20)(15,24,38,58)(17,32)(18,28)(19,29)(22,41)(23,46,55,50)(26,61,60,68)(27,62)(31,45)(33,64)(35,42)(36,63)(37,54,40,67)(39,51,66,71)(43,53)(44,52)(47,69,65,59)(49,57,70,56);(8,10,22,51,13,26,12,24,29,55,32,59,31,57,63,67,14,25,53,71,35,60,33,58,11,23,52,69,28,56,27,54)(9,15,36,61,18,39,17,21,16,37,30,49,42,47,41,46,20,38,62,68,45,66,44,48,19,40,64,70,34,65,43,50);(8,11,14,29)(9,16,20,19)(10,23,25,55)(12,27,33,63)(13,28,35,31)(15,37,38,40)(17,41,44,43)(18,42,45,34)(21,46,48,50)(22,52,53,32)(24,54,58,67)(26,56,60,57)(30,62,64,36)(39,47,66,65)(49,68,70,61)(51,69,71,59)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6626);