Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#99 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C2×Q8 | C24×C7 |
| prime radical | C7×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×D4 ◁ C2×C7×2−1+4 ◁ G | |
1176 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)(8,9,13,18,14,19,10,20)(11,16,25,32,23,30,21,34)(12,17,22,29,27,33,26,15)(24,31,35,38,36,39,37,28);(8,10,14,13)(9,15,19,29)(11,21,23,25)(12,26,27,22)(16,28,30,38)(17,18,33,20)(24,37,36,35)(31,32,39,34);(8,11)(9,16)(10,21)(12,24)(13,25)(14,23)(15,28)(17,31)(18,32)(19,30)(20,34)(22,35)(26,37)(27,36)(29,38)(33,39);(11,23)(16,30)(21,25)(24,36)(28,38)(31,39)(32,34)(35,37)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26360);