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G = C2xD40order 160 = 25·5

Direct product of C2 and D40

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

Aliases: C2xD40, C8:7D10, C10:1D8, C4.7D20, C40:8C22, C20.30D4, D20:3C22, C20.29C23, C22.13D20, C5:1(C2xD8), (C2xC8):3D5, (C2xC40):5C2, (C2xD20):5C2, C2.12(C2xD20), C10.10(C2xD4), (C2xC4).80D10, (C2xC10).17D4, C4.27(C22xD5), (C2xC20).89C22, SmallGroup(160,124)

Series: Derived Chief Lower central Upper central

C1C20 — C2xD40
C1C5C10C20D20C2xD20 — C2xD40
C5C10C20 — C2xD40
C1C22C2xC4C2xC8

Generators and relations for C2xD40
 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, C2xC4, D4, C23, D5, C10, C10, C2xC8, D8, C2xD4, C20, D10, C2xC10, C2xD8, C40, D20, D20, C2xC20, C22xD5, D40, C2xC40, C2xD20, C2xD40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2xD4, D10, C2xD8, D20, C22xD5, D40, C2xD20, C2xD40

Smallest permutation representation of C2xD40
On 80 points
Generators in S80
(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)]])

C2xD40 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  C2xD5xD8  D8:15D10
C2xD40 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 2A2B2C2D2E2F2G4A4B5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122222224455888810···1020···2040···40
size111120202020222222222···22···22···2

46 irreducible representations

dim1111222222222
type+++++++++++++
imageC1C2C2C2D4D4D5D8D10D10D20D20D40
kernelC2xD40D40C2xC40C2xD20C20C2xC10C2xC8C10C8C2xC4C4C22C2
# reps14121124424416

Matrix representation of C2xD40 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] >;

C2xD40 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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