Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C2×C7×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
2650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2)(35,36,37,38,39,40,41);(3,4)(5,15)(6,20)(7,12)(8,13)(9,14)(10,17)(11,21)(16,30)(18,26)(19,25)(22,27)(23,28)(24,32)(29,31)(33,34);(3,5,7,17,9,19,8,18)(4,10,13,15,14,26,12,25)(6,16,21,31,22,32,23,33)(11,24,20,29,28,30,27,34);(3,6)(4,11)(5,16)(7,21)(8,23)(9,22)(10,24)(12,27)(13,20)(14,28)(15,29)(17,31)(18,33)(19,32)(25,34)(26,30);(3,7,9,8)(4,12,14,13)(5,17,19,18)(6,21,22,23)(10,25,26,15)(11,27,28,20)(16,31,32,33)(24,34,30,29)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53454);