Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13335 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2011 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
536 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)(10,20)(11,25)(12,30)(13,36)(14,18)(15,22)(16,27)(17,34)(19,33)(21,39)(23,61)(24,53)(26,44)(28,66)(29,43)(31,49)(32,50)(35,51)(37,46)(38,47)(40,60)(41,67)(42,57)(45,65)(48,58)(52,55)(54,63)(56,71)(59,69)(62,70)(64,68);(8,10,14,24)(9,15,18,40)(11,21,29,57)(12,31,35,67)(13,37,33,23)(16,42,44,39)(17,46,48,61)(19,49,36,41)(20,51,53,30)(22,58,60,34)(25,63,43,52)(26,64,27,59)(28,62,38,56)(32,55,65,54)(45,70,50,71)(47,68,66,69);(8,11)(9,16)(10,21)(12,32)(13,38)(14,29)(15,39)(17,47)(18,44)(19,50)(20,52)(22,59)(23,62)(24,57)(25,30)(26,58)(27,34)(28,33)(31,55)(35,65)(36,45)(37,56)(40,42)(41,71)(43,51)(46,69)(48,66)(49,70)(53,63)(54,67)(60,64)(61,68)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6323);