Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60389 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×#41 ◁ G | |
4770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,17)(10,18,21,31)(11,15,22,29)(12,24,26,16)(14,25,28,20)(19,32,33,39)(23,36,35,30)(27,37,38,34);(8,10)(9,14)(11,19)(12,25)(13,21)(15,27)(16,31)(17,28)(18,24)(20,26)(22,33)(23,37)(29,38)(30,39)(32,36)(34,35);(12,26)(16,24)(18,31)(20,25)(23,35)(30,36)(32,39)(34,37);(8,11)(9,15)(10,19)(12,23)(13,22)(14,27)(16,30)(17,29)(18,32)(20,34)(21,33)(24,36)(25,37)(26,35)(28,38)(31,39);(11,22)(15,29)(19,33)(23,35)(27,38)(30,36)(32,39)(34,37)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53377);