Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13323 |
| 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 | |
534 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)(9,15)(10,20)(11,24)(12,13)(16,33)(17,31)(18,25)(19,32)(21,34)(22,26)(23,29)(27,28)(30,35)(36,39);(8,9)(10,21)(11,25)(12,17)(13,18)(14,19)(15,24)(16,20)(22,30)(23,35)(26,34)(27,32)(28,31)(29,36)(33,37)(38,39);(8,10)(9,16)(11,26)(12,29)(13,20)(14,23)(15,30)(17,21)(18,35)(19,36)(22,28)(24,38)(25,39)(27,37)(31,33)(32,34)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6311);