Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1807 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,21,25,39)(11,26,28,36)(12,22)(13,18)(14,20)(15,23,29,16)(17,30)(19,32)(24,38,33,27)(31,34)(35,37);(8,10,12,23,13,24,30,36,14,25,31,16,32,33,37,26)(9,15,17,21,19,11,22,27,20,29,35,39,18,28,34,38);(8,11)(9,16)(10,22)(12,27)(13,29)(14,28)(15,32)(17,33)(18,36)(19,26)(20,23)(21,37)(24,35)(25,34)(30,39)(31,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5821);