Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#23412 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
1948 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)(10,20)(11,25)(12,17)(13,18)(14,19)(15,23)(16,26)(21,29)(22,30)(24,34)(27,33)(28,32)(31,37)(35,36)(38,39);(8,10)(9,15)(11,21)(12,22)(13,23)(14,24)(16,31)(17,20)(18,34)(19,30)(25,35)(26,36)(27,38)(28,37)(29,33)(32,39);(8,11)(9,16)(10,21)(12,26)(13,27)(14,28)(15,29)(17,32)(18,25)(19,33)(20,35)(22,36)(23,38)(24,37)(30,31)(34,39);(8,12,14,13)(9,17,19,18)(10,22,24,23)(11,26,28,27)(15,20,30,34)(16,32,33,25)(21,36,37,38)(29,35,31,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16400);