Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,21)(11,25)(12,20)(13,18)(14,19)(15,22)(16,23)(17,29)(24,37)(26,38)(27,33)(28,34)(30,36)(31,35)(32,39);(8,10)(9,15)(11,26)(12,23)(13,31)(14,24)(16,34)(17,35)(18,32)(19,33)(20,38)(21,25)(22,29)(27,28)(30,39)(36,37);(8,10)(9,16)(11,27)(12,31)(13,23)(14,24)(15,17)(18,37)(19,35)(20,21)(22,30)(25,32)(26,28)(29,39)(33,34)(36,38)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6306);