Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×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)(8,9,15,21)(10,22,27,57)(11,28,34,64)(12,19,37,54)(13,18,38,53)(14,39,36,20)(16,41,45,24)(17,46,51,31)(23,58,62,42)(25,44,35,55)(26,43,40,52)(29,65,59,70)(30,50,32,68)(33,48,67,49)(47,60,56,69)(61,63,66,71);(8,10)(9,16)(11,29)(12,35)(13,24)(14,40)(15,27)(17,47)(18,52)(19,22)(20,55)(21,45)(23,32)(25,37)(26,36)(28,63)(30,62)(31,61)(33,69)(34,59)(38,41)(39,44)(42,49)(43,53)(46,66)(48,58)(50,65)(51,56)(54,57)(60,67)(64,71)(68,70);(8,11,12,30,36,67,38,46,15,34,37,32,14,33,13,31)(9,17,18,48,39,68,54,64,21,51,53,49,20,50,19,28)(10,23,24,59,40,66,25,60,27,62,41,29,26,61,35,69)(16,42,22,56,55,71,43,70,45,58,57,47,44,63,52,65)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6431);