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×#13339 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
534 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)(14,26,31,61)(15,32,38,51)(16,39,44,57)(17,24,18,23)(20,45,49,28)(21,34,54,35)(22,42,58,43)(27,46,65,74)(29,55,30,60)(33,70,64,50)(36,56,37,69)(40,47,67,48)(41,52,72,53)(59,62,75,63)(66,68,71,73);(8,9)(10,11)(12,14,19,31)(13,20,25,49)(15,33,38,64)(16,40,44,67)(17,30,18,29)(21,50,54,70)(22,55,58,60)(23,48,24,47)(26,57,61,39)(27,37,65,36)(28,42,45,43)(32,68,51,73)(34,62,35,63)(41,71,72,66)(46,53,74,52)(56,75,69,59);(9,11)(12,15)(13,21)(14,27)(16,41)(17,37)(18,36)(19,38)(20,46)(22,56)(23,53)(24,52)(25,54)(26,59)(28,66)(29,64)(30,33)(31,65)(32,39)(34,42)(35,43)(40,63)(44,72)(45,71)(47,70)(48,50)(49,74)(51,57)(55,68)(58,69)(60,73)(61,75)(62,67)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6327);