Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 75 points:
(1,2,3,4,5,6,7)(9,11)(12,13,18,25,42,58,43,24,20,27,16,23,19,26,17,44)(14,28,32,51,46,49,31,21,34,50,30,60,33,57,45,52)(15,35,38,56,47,53,41,55,40,59,36,22,39,54,37,61)(29,63,66,62,70,75,65,71,68,74,64,72,67,48,69,73);(8,9,10,11)(12,14)(13,21)(15,29)(16,31)(17,32)(18,45)(19,46)(20,34)(22,48)(23,57)(24,28)(25,49)(26,60)(27,52)(30,43)(33,42)(35,71)(36,65)(37,66)(38,69)(39,70)(40,68)(41,64)(44,50)(47,67)(51,58)(53,62)(54,72)(55,63)(56,75)(59,73)(61,74);(12,15,20,40)(13,22,27,56)(14,29,34,68)(16,41,18,37)(17,38,43,36)(19,47,42,39)(21,48,52,75)(23,35,25,59)(24,53,44,54)(26,61,58,55)(28,62,50,72)(30,65,32,69)(31,64,45,66)(33,70,46,67)(49,73,57,71)(51,63,60,74)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6008);