Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#9715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#2856 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,16,18,19)(15,17)(20,21,22,25,23,26,27,24);(12,14,15,18)(13,16)(17,19)(21,24)(22,27)(25,26);(8,9,10,11)(14,18)(16,19)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2703);