Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#17991 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C23×C7 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C7×#431 ◁ G | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7)(8,9,12,16,13,15,21,22)(10,14,18,11,17,19,23,20)(24,25)(26,34)(27,31)(28,40)(29,32)(30,36)(33,42)(35,48)(37,47)(38,46)(39,52)(41,44)(43,50)(45,51)(49,55)(53,54);(8,10)(9,14)(11,16)(12,18)(13,17)(15,19)(20,22)(21,23)(24,26,29,37)(25,30,32,44)(27,35,38,49)(28,41,42,36)(31,43,46,54)(33,47,40,34)(39,53,51,50)(45,55,52,48);(8,11,12,19,13,20,21,14)(9,10,16,18,15,17,22,23)(24,27,29,38)(25,31,32,46)(26,35,37,49)(28,39,42,51)(30,43,44,54)(33,45,40,52)(34,48,47,55)(36,50,41,53);(9,15)(10,17)(16,22)(18,23)(25,32)(27,38)(30,44)(33,40)(34,47)(35,49)(39,51)(50,53)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10979);