Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#10779 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #379 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(12,13)(14,23)(15,27,29,36)(16,30)(17,21)(18,22)(19,25)(20,28,37,32)(24,39,42,33)(26,34)(31,35)(38,41,43,40);(8,9,10,11)(12,14)(13,19)(15,24)(16,31)(17,25)(18,26)(20,33)(21,34)(22,35)(23,30)(27,38)(28,41)(29,42)(32,40)(36,43)(37,39);(12,15)(13,20)(14,24)(16,28)(17,32)(18,29)(19,33)(21,36)(22,37)(23,38)(25,40)(26,42)(27,30)(31,41)(34,43)(35,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3767);