Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13322 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 75 points:
(1,2,3,4,5,6,7)(12,13,19,25)(14,26,31,57)(15,32,35,54)(16,36,41,52)(17,24,18,23)(20,43,47,28)(21,42,49,34)(22,39,53,40)(27,58,61,75)(29,50,30,56)(33,51,38,48)(37,45,63,46)(44,67,74,70)(55,64,72,60)(59,65,62,73)(66,71,69,68);(8,9)(10,11)(12,14,19,31)(13,20,25,47)(15,27,35,61)(16,37,41,63)(17,30,18,29)(21,44,49,74)(22,50,53,56)(23,46,24,45)(26,52,57,36)(28,39,43,40)(32,65,54,73)(33,67,38,70)(34,64,42,60)(48,75,51,58)(55,66,72,69)(59,71,62,68);(9,11)(12,15)(13,21)(14,27)(16,38)(17,42)(18,34)(19,35)(20,44)(22,51)(23,54)(24,32)(25,49)(26,55)(28,62)(29,64)(30,60)(31,61)(33,41)(36,69)(37,70)(39,68)(40,71)(43,59)(45,65)(46,73)(47,74)(48,53)(50,58)(52,66)(56,75)(57,72)(63,67)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6310);