Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#7537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D32 |
| commutator | C32 | C2×C4×C7 |
| Frattini, cosocle | C2×C32 | C22×C7 |
| prime radical | C2×C7×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C2×C7×C16 ◁ C2×C7×C32 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,19)(15,30)(16,34)(17,37)(18,24)(20,25)(21,26)(22,31)(23,35)(27,47)(28,49)(29,41)(32,65)(33,63)(36,67)(38,51)(39,50)(40,58)(42,57)(43,60)(44,53)(45,66)(46,56)(48,64)(52,68)(54,62)(55,61)(59,72)(69,75)(70,74)(71,73);(13,19)(14,25)(15,26)(16,31)(17,35)(20,21)(22,47)(23,49)(24,41)(27,57)(28,60)(29,53)(30,38)(32,66)(33,56)(34,39)(36,64)(37,40)(42,62)(43,61)(44,46)(45,68)(48,72)(50,51)(52,58)(54,55)(59,75)(63,70)(65,69)(67,71)(73,74)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 525);