Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#21000 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#121 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#386 |
| prime radical | C2×C7×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
1704 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)(10,14,19,28)(11,21,22,33)(12,17)(13,16)(15,23,30,25)(18,34)(20,29,24,32)(26,31)(27,37)(35,38)(36,39);(8,10)(9,14,16,28)(11,18)(12,24)(13,19)(15,27,30,39)(17,32,31,29)(20,26)(21,38,33,34)(22,35)(23,37)(25,36);(8,11)(9,15)(10,18)(12,25)(13,22)(14,27)(16,30)(17,33)(19,35)(20,37)(21,31)(23,26)(24,36)(28,39)(29,38)(32,34);(10,19)(14,28)(18,35)(20,24)(27,39)(29,32)(34,38)(36,37)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13988);