Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#23873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,22)(12,25)(13,17)(14,20)(15,23)(16,27)(19,30)(21,32)(24,33)(26,29)(28,31)(34,35)(36,38)(37,39);(8,10,13,21)(9,14,17,29)(11,19,24,36)(12,26,27,20)(15,30,31,38)(16,32,25,18)(22,34,33,39)(23,37,28,35);(12,27)(14,29)(16,25)(20,26)(22,33)(23,28)(30,38)(35,37);(8,11)(9,15)(10,19)(12,28)(13,24)(14,30)(16,33)(17,31)(18,34)(20,37)(21,36)(22,25)(23,27)(26,35)(29,38)(32,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16861);