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G = C2×D40order 160 = 25·5

Direct product of C2 and D40

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D40, C87D10, C101D8, C4.7D20, C408C22, C20.30D4, D203C22, C20.29C23, C22.13D20, C51(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

C1C20 — C2×D40
C1C5C10C20D20C2×D20 — C2×D40
C5C10C20 — C2×D40
C1C22C2×C4C2×C8

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 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C5, C8 [×2], C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], D10 [×8], C2×C10, C2×D8, C40 [×2], D20 [×4], D20 [×2], C2×C20, C22×D5 [×2], D40 [×4], C2×C40, C2×D20 [×2], C2×D40
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, D20 [×2], C22×D5, D40 [×2], C2×D20, C2×D40

Smallest permutation representation of C2×D40
On 80 points
Generators in S80
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)
(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 47)(42 46)(43 45)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)

G:=sub<Sym(80)| (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71), (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)(41,47)(42,46)(43,45)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)>;

G:=Group( (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71), (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)(41,47)(42,46)(43,45)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65) );

G=PermutationGroup([(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71)], [(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),(41,47),(42,46),(43,45),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65)])

C2×D40 is a maximal subgroup of
C40.5D4  D40.6C4  D407C4  D40.4C4  C204D8  C8.8D20  D409C4  C8⋊D20  D2013D4  D2014D4  D4⋊D20  D203D4  D204D4  D20.12D4  C4⋊D40  D20.19D4  C82D20  D4015C4  D4012C4  C87D20  C8.21D20  D80⋊C2  C4029D4  C403D4  D4.4D20  C405D4  C409D4  C40.28D4  D8⋊D10  D4.12D20  C2×D5×D8  D815D10
C2×D40 is a maximal quotient of
C408Q8  C4.5D40  C204D8  D2013D4  C22.D40  C4⋊D40  D204Q8  D807C2  D80⋊C2  C16.D10  C4029D4

46 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122222224455888810···1020···2040···40
size111120202020222222222···22···22···2

46 irreducible representations

dim1111222222222
type+++++++++++++
imageC1C2C2C2D4D4D5D8D10D10D20D20D40
kernelC2×D40D40C2×C40C2×D20C20C2×C10C2×C8C10C8C2×C4C4C22C2
# reps14121124424416

Matrix representation of C2×D40 in GL3(𝔽41) generated by

4000
010
001
,
100
02338
035
,
4000
01625
03925
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

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