Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#11461 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C8 |
| prime radical | C24×C7 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,16,18,13)(11,14)(12,15)(17,22,23,20)(19,21)(24,25,27,31,28,32,37,43,29,33,38,44,39,45,49,53)(26,30,35,41,36,42,48,52,34,40,46,50,47,51,54,55);(8,10)(9,13)(11,17)(12,18)(14,20)(15,16)(19,23)(21,22)(24,26)(25,30)(27,35)(28,36)(29,34)(31,41)(32,42)(33,40)(37,48)(38,46)(39,47)(43,52)(44,50)(45,51)(49,54)(53,55);(8,11,12,19)(9,14,15,21)(10,17,18,23)(13,20,16,22)(26,34)(30,40)(35,46)(36,47)(41,50)(42,51)(48,54)(52,55)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4449);