Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#7980 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #95 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,15)(12,16)(13,17)(14,19)(20,23)(21,22)(24,25)(26,32,34,29)(27,30)(28,31)(33,38,39,36)(35,37);(8,10,12,20,13,21,11,19)(9,14,15,22,17,23,16,18)(24,26)(25,29)(27,33)(28,34)(30,36)(31,32)(35,39)(37,38);(8,11,13,12)(9,15,17,16)(10,19,21,20)(14,22,23,18)(24,27,28,35)(25,30,31,37)(26,33,34,39)(29,36,32,38)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 968);