direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D40, C8⋊7D10, C10⋊1D8, C4.7D20, C40⋊8C22, C20.30D4, D20⋊3C22, C20.29C23, C22.13D20, C5⋊1(C2×D8), (C2×C8)⋊3D5, (C2×C40)⋊5C2, (C2×D20)⋊5C2, C2.12(C2×D20), C10.10(C2×D4), (C2×C4).80D10, (C2×C10).17D4, C4.27(C22×D5), (C2×C20).89C22, SmallGroup(160,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D40
G = < a,b,c | a2=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 376 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, C23, D5, C10, C10, C2×C8, D8, C2×D4, C20, D10, C2×C10, C2×D8, C40, D20, D20, C2×C20, C22×D5, D40, C2×C40, C2×D20, C2×D40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, D20, C22×D5, D40, C2×D20, C2×D40
►On 80 points
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 51)(42 50)(43 49)(44 48)(45 47)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
G:=sub<Sym(80)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67) );
G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,51),(42,50),(43,49),(44,48),(45,47),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)]])
C2×D40 is a maximal subgroup of
C40.5D4 D40.6C4 D40⋊7C4 D40.4C4 C20⋊4D8 C8.8D20 D40⋊9C4 C8⋊D20 D20⋊13D4 D20⋊14D4 D4⋊D20 D20⋊3D4 D20⋊4D4 D20.12D4 C4⋊D40 D20.19D4 C8⋊2D20 D40⋊15C4 D40⋊12C4 C8⋊7D20 C8.21D20 D80⋊C2 C40⋊29D4 C40⋊3D4 D4.4D20 C40⋊5D4 C40⋊9D4 C40.28D4 D8⋊D10 D4.12D20 C2×D5×D8 D8⋊15D10
C2×D40 is a maximal quotient of
C40⋊8Q8 C4.5D40 C20⋊4D8 D20⋊13D4 C22.D40 C4⋊D40 D20⋊4Q8 D80⋊7C2 D80⋊C2 C16.D10 C40⋊29D4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D10 | D10 | D20 | D20 | D40 |
| kernel | C2×D40 | D40 | C2×C40 | C2×D20 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C2×D40 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 23 | 38 |
| 0 | 3 | 5 |
| 40 | 0 | 0 |
| 0 | 16 | 25 |
| 0 | 39 | 25 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,23,3,0,38,5],[40,0,0,0,16,39,0,25,25] >;
C2×D40 in GAP, Magma, Sage, TeX
C_2\times D_{40} % in TeX
G:=Group("C2xD40"); // GroupNames label
G:=SmallGroup(160,124);
// by ID
G=gap.SmallGroup(160,124);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,122,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations