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G = C10×D8order 160 = 25·5

Direct product of C10 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C10×D8, C20.41D4, C4012C22, C20.44C23, (C2×C8)⋊3C10, C82(C2×C10), C4.6(C5×D4), (C2×C40)⋊11C2, D41(C2×C10), (C2×D4)⋊4C10, (D4×C10)⋊13C2, C10.74(C2×D4), C2.11(D4×C10), (C2×C10).52D4, (C5×D4)⋊10C22, C4.1(C22×C10), C22.14(C5×D4), (C2×C20).129C22, (C2×C4).25(C2×C10), SmallGroup(160,193)

Series: Derived Chief Lower central Upper central

C1C4 — C10×D8
C1C2C4C20C5×D4C5×D8 — C10×D8
C1C2C4 — C10×D8
C1C2×C10C2×C20 — C10×D8

Generators and relations for C10×D8
 G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C5, C8 [×2], C2×C4, D4 [×4], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×4], C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], C2×C10, C2×C10 [×8], C2×D8, C40 [×2], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], C2×C40, C5×D8 [×4], D4×C10 [×2], C10×D8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], D8 [×2], C2×D4, C2×C10 [×7], C2×D8, C5×D4 [×2], C22×C10, C5×D8 [×2], D4×C10, C10×D8

Smallest permutation representation of C10×D8
On 80 points
Generators in S80
(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 28 50 17 52 37 62 79)(2 29 41 18 53 38 63 80)(3 30 42 19 54 39 64 71)(4 21 43 20 55 40 65 72)(5 22 44 11 56 31 66 73)(6 23 45 12 57 32 67 74)(7 24 46 13 58 33 68 75)(8 25 47 14 59 34 69 76)(9 26 48 15 60 35 70 77)(10 27 49 16 51 36 61 78)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 61)(28 62)(29 63)(30 64)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 41)(39 42)(40 43)

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

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

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

C10×D8 is a maximal subgroup of
C10.D16  D8.Dic5  D8.D10  Dic5⋊D8  C405D4  D8⋊Dic5  (C2×D8).D5  C4011D4  C40.22D4  D20⋊D4  C406D4  Dic10⋊D4  C4012D4  C40.23D4  D813D10

70 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D8A8B8C8D10A···10L10M···10AB20A···20H40A···40P
order12222222445555888810···1010···1020···2040···40
size1111444422111122221···14···42···22···2

70 irreducible representations

dim11111111222222
type+++++++
imageC1C2C2C2C5C10C10C10D4D4D8C5×D4C5×D4C5×D8
kernelC10×D8C2×C40C5×D8D4×C10C2×D8C2×C8D8C2×D4C20C2×C10C10C4C22C2
# reps1142441681144416

Matrix representation of C10×D8 in GL3(𝔽41) generated by

4000
0250
0025
,
4000
0029
02417
,
100
01712
01724
G:=sub<GL(3,GF(41))| [40,0,0,0,25,0,0,0,25],[40,0,0,0,0,24,0,29,17],[1,0,0,0,17,17,0,12,24] >;

C10×D8 in GAP, Magma, Sage, TeX

C_{10}\times D_8
% in TeX

G:=Group("C10xD8");
// GroupNames label

G:=SmallGroup(160,193);
// by ID

G=gap.SmallGroup(160,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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