Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#33690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24×C7 |
| prime radical | C7×#2190 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×M16 ◁ G | |
526 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)(10,15)(11,21)(12,25)(13,29)(14,18)(16,24)(17,26)(19,36)(20,35)(22,34)(23,38)(27,30)(28,33)(31,37)(32,39);(8,10,14,13)(9,15,18,29)(11,19,23,22)(12,26,28,20)(16,30,31,32)(17,33,35,25)(21,34,38,36)(24,39,37,27);(8,11)(9,16)(10,19)(12,27)(13,22)(14,23)(15,30)(17,34)(18,31)(20,37)(21,25)(24,26)(28,39)(29,32)(33,38)(35,36);(11,22,23,19)(12,28)(16,30,31,32)(17,35)(20,26)(21,34,38,36)(24,27,37,39)(25,33)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26678);