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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C10×D8
 Chief series C1 — C2 — C4 — C20 — C5×D4 — C5×D8 — C10×D8
 Lower central C1 — C2 — C4 — C10×D8
 Upper central C1 — C2×C10 — C2×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, C2, C4, C22, C22, C5, C8, C2×C4, D4, D4, C23, C10, C10, C10, C2×C8, D8, C2×D4, C20, C2×C10, C2×C10, C2×D8, C40, C2×C20, C5×D4, C5×D4, C22×C10, C2×C40, C5×D8, D4×C10, C10×D8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, C5×D8, 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 32 70 20 56 79)(2 29 41 33 61 11 57 80)(3 30 42 34 62 12 58 71)(4 21 43 35 63 13 59 72)(5 22 44 36 64 14 60 73)(6 23 45 37 65 15 51 74)(7 24 46 38 66 16 52 75)(8 25 47 39 67 17 53 76)(9 26 48 40 68 18 54 77)(10 27 49 31 69 19 55 78)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 69)(32 70)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)

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

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

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

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 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10AB 20A ··· 20H 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 4 4 4 4 2 2 1 1 1 1 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D4 D8 C5×D4 C5×D4 C5×D8 kernel C10×D8 C2×C40 C5×D8 D4×C10 C2×D8 C2×C8 D8 C2×D4 C20 C2×C10 C10 C4 C22 C2 # reps 1 1 4 2 4 4 16 8 1 1 4 4 4 16

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

 40 0 0 0 25 0 0 0 25
,
 40 0 0 0 0 29 0 24 17
,
 1 0 0 0 17 12 0 17 24
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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