Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
520 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)(10,22)(11,28)(12,32)(13,21)(14,19)(15,20)(16,24)(17,30)(18,35)(23,52)(25,45)(26,50)(27,51)(29,61)(31,48)(33,39)(34,47)(36,46)(37,44)(38,62)(40,57)(41,56)(42,67)(43,64)(49,54)(53,59)(55,69)(58,68)(60,63)(65,70)(66,71);(8,10,15,27)(9,16,20,40)(11,23,31,55)(12,33,36,56)(13,26,14,25)(17,42,44,38)(18,45,47,50)(19,41,21,39)(22,46,51,32)(24,34,57,35)(28,59,48,66)(29,62,64,67)(30,58,37,54)(43,69,61,52)(49,60,68,70)(53,65,71,63);(8,11)(9,17)(10,23)(12,29)(13,37)(14,30)(15,31)(16,38)(18,43)(19,48)(20,44)(21,28)(22,49)(24,53)(25,58)(26,54)(27,55)(32,60)(33,62)(34,65)(35,63)(36,64)(39,66)(40,42)(41,59)(45,52)(46,70)(47,61)(50,69)(51,68)(56,67)(57,71)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6299);