Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9050 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #386 |
| prime radical | C22×C7×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,12,15,13,10,11,14)(16,17,20,24,21,18,22,26)(19,25,28,31,27,23,29,30);(8,10,13,9)(11,14,12,15)(16,18,20,26,21,17,22,24)(19,25,29,30,27,23,28,31);(8,9,13,10)(11,15,12,14)(16,19)(17,23)(18,25)(20,29)(21,27)(22,28)(24,31)(26,30)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2038);