Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#11855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
324 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)(17,22,19,21)(20,23)(24,25)(26,36)(27,33)(28,43)(29,46)(30,35)(31,38)(32,41)(34,45)(37,49)(39,55)(40,51)(42,52)(44,53)(47,48)(50,54);(8,10)(9,13)(11,19)(12,18)(14,22)(15,16)(17,20)(21,23)(24,26)(25,31)(27,38)(28,44)(29,47)(30,40)(32,51)(33,48)(34,53)(35,50)(36,43)(37,41)(39,45)(42,54)(46,49)(52,55);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,27,29,41,30,42,45,28)(25,32,34,33,35,43,46,52)(26,37,39,38,40,44,47,54)(31,48,49,51,50,55,53,36)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4843);