Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#12767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1790 |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | C22×D4 | C23×C7 |
| prime radical | C7×#3543 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,17,19,14)(11,15,21,22)(12,16)(13,18)(20,23);(8,10)(9,13)(11,18)(12,19)(14,22)(15,17)(16,23)(20,21);(8,10)(9,14)(11,20)(12,19)(13,22)(15,23)(16,17)(18,21)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5755);