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) = C24 |
| direct product | C7×#23236 |
| 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×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×C4⋊C8 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,13)(14,15)(16,17,22,27,21,26,20,25)(18,24,28,31,29,23,19,30);(8,9)(10,14)(11,15)(12,13)(16,18,22,28,21,29,20,19)(17,23,27,30,26,24,25,31);(8,10,12,11)(9,14,13,15)(16,19,20,29,21,28,22,18)(17,24,27,31,26,23,25,30);(8,11,12,10)(9,14,13,15)(16,20,21,22)(17,25,26,27)(18,28,29,19)(23,30,24,31)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16224);