Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#23115 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C7×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×C4⋊C8 ◁ G | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7)(9,13)(14,15)(16,17,21,26,22,25,20,34)(18,23,29,38,30,37,28,44)(19,31,32,24,33,40,35,39)(27,41,42,36,43,47,45,46);(8,9,12,13)(10,14,11,15)(16,18,21,29,22,30,20,28)(17,23,26,38,25,37,34,44)(19,27,32,42,33,43,35,45)(24,36,40,47,39,46,31,41);(8,10,12,11)(9,14,13,15)(16,19)(17,24)(18,27)(20,35)(21,32)(22,33)(23,36)(25,39)(26,40)(28,45)(29,42)(30,43)(31,34)(37,46)(38,47)(41,44);(8,11,12,10)(9,14,13,15)(17,25)(23,37)(24,39)(26,34)(31,40)(36,46)(38,44)(41,47)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16103);