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## G = C10×Q16order 160 = 25·5

### Direct product of C10 and Q16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C10×Q16
 Chief series C1 — C2 — C4 — C20 — C5×Q8 — C5×Q16 — C10×Q16
 Lower central C1 — C2 — C4 — C10×Q16
 Upper central C1 — C2×C10 — C2×C20 — C10×Q16

Generators and relations for C10×Q16
G = < a,b,c | a10=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C10, C2×C8, Q16, C2×Q8, C20, C20, C2×C10, C2×Q16, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C2×C40, C5×Q16, Q8×C10, C10×Q16
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C2×C10, C2×Q16, C5×D4, C22×C10, C5×Q16, D4×C10, C10×Q16

Smallest permutation representation of C10×Q16
Regular action on 160 points
Generators in S160
(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)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 23 50 154 52 32 62 12)(2 24 41 155 53 33 63 13)(3 25 42 156 54 34 64 14)(4 26 43 157 55 35 65 15)(5 27 44 158 56 36 66 16)(6 28 45 159 57 37 67 17)(7 29 46 160 58 38 68 18)(8 30 47 151 59 39 69 19)(9 21 48 152 60 40 70 20)(10 22 49 153 51 31 61 11)(71 149 87 111 95 127 109 135)(72 150 88 112 96 128 110 136)(73 141 89 113 97 129 101 137)(74 142 90 114 98 130 102 138)(75 143 81 115 99 121 103 139)(76 144 82 116 100 122 104 140)(77 145 83 117 91 123 105 131)(78 146 84 118 92 124 106 132)(79 147 85 119 93 125 107 133)(80 148 86 120 94 126 108 134)
(1 93 52 79)(2 94 53 80)(3 95 54 71)(4 96 55 72)(5 97 56 73)(6 98 57 74)(7 99 58 75)(8 100 59 76)(9 91 60 77)(10 92 51 78)(11 124 153 146)(12 125 154 147)(13 126 155 148)(14 127 156 149)(15 128 157 150)(16 129 158 141)(17 130 159 142)(18 121 160 143)(19 122 151 144)(20 123 152 145)(21 117 40 131)(22 118 31 132)(23 119 32 133)(24 120 33 134)(25 111 34 135)(26 112 35 136)(27 113 36 137)(28 114 37 138)(29 115 38 139)(30 116 39 140)(41 86 63 108)(42 87 64 109)(43 88 65 110)(44 89 66 101)(45 90 67 102)(46 81 68 103)(47 82 69 104)(48 83 70 105)(49 84 61 106)(50 85 62 107)

G:=sub<Sym(160)| (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,23,50,154,52,32,62,12)(2,24,41,155,53,33,63,13)(3,25,42,156,54,34,64,14)(4,26,43,157,55,35,65,15)(5,27,44,158,56,36,66,16)(6,28,45,159,57,37,67,17)(7,29,46,160,58,38,68,18)(8,30,47,151,59,39,69,19)(9,21,48,152,60,40,70,20)(10,22,49,153,51,31,61,11)(71,149,87,111,95,127,109,135)(72,150,88,112,96,128,110,136)(73,141,89,113,97,129,101,137)(74,142,90,114,98,130,102,138)(75,143,81,115,99,121,103,139)(76,144,82,116,100,122,104,140)(77,145,83,117,91,123,105,131)(78,146,84,118,92,124,106,132)(79,147,85,119,93,125,107,133)(80,148,86,120,94,126,108,134), (1,93,52,79)(2,94,53,80)(3,95,54,71)(4,96,55,72)(5,97,56,73)(6,98,57,74)(7,99,58,75)(8,100,59,76)(9,91,60,77)(10,92,51,78)(11,124,153,146)(12,125,154,147)(13,126,155,148)(14,127,156,149)(15,128,157,150)(16,129,158,141)(17,130,159,142)(18,121,160,143)(19,122,151,144)(20,123,152,145)(21,117,40,131)(22,118,31,132)(23,119,32,133)(24,120,33,134)(25,111,34,135)(26,112,35,136)(27,113,36,137)(28,114,37,138)(29,115,38,139)(30,116,39,140)(41,86,63,108)(42,87,64,109)(43,88,65,110)(44,89,66,101)(45,90,67,102)(46,81,68,103)(47,82,69,104)(48,83,70,105)(49,84,61,106)(50,85,62,107)>;

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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,23,50,154,52,32,62,12)(2,24,41,155,53,33,63,13)(3,25,42,156,54,34,64,14)(4,26,43,157,55,35,65,15)(5,27,44,158,56,36,66,16)(6,28,45,159,57,37,67,17)(7,29,46,160,58,38,68,18)(8,30,47,151,59,39,69,19)(9,21,48,152,60,40,70,20)(10,22,49,153,51,31,61,11)(71,149,87,111,95,127,109,135)(72,150,88,112,96,128,110,136)(73,141,89,113,97,129,101,137)(74,142,90,114,98,130,102,138)(75,143,81,115,99,121,103,139)(76,144,82,116,100,122,104,140)(77,145,83,117,91,123,105,131)(78,146,84,118,92,124,106,132)(79,147,85,119,93,125,107,133)(80,148,86,120,94,126,108,134), (1,93,52,79)(2,94,53,80)(3,95,54,71)(4,96,55,72)(5,97,56,73)(6,98,57,74)(7,99,58,75)(8,100,59,76)(9,91,60,77)(10,92,51,78)(11,124,153,146)(12,125,154,147)(13,126,155,148)(14,127,156,149)(15,128,157,150)(16,129,158,141)(17,130,159,142)(18,121,160,143)(19,122,151,144)(20,123,152,145)(21,117,40,131)(22,118,31,132)(23,119,32,133)(24,120,33,134)(25,111,34,135)(26,112,35,136)(27,113,36,137)(28,114,37,138)(29,115,38,139)(30,116,39,140)(41,86,63,108)(42,87,64,109)(43,88,65,110)(44,89,66,101)(45,90,67,102)(46,81,68,103)(47,82,69,104)(48,83,70,105)(49,84,61,106)(50,85,62,107) );

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),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,23,50,154,52,32,62,12),(2,24,41,155,53,33,63,13),(3,25,42,156,54,34,64,14),(4,26,43,157,55,35,65,15),(5,27,44,158,56,36,66,16),(6,28,45,159,57,37,67,17),(7,29,46,160,58,38,68,18),(8,30,47,151,59,39,69,19),(9,21,48,152,60,40,70,20),(10,22,49,153,51,31,61,11),(71,149,87,111,95,127,109,135),(72,150,88,112,96,128,110,136),(73,141,89,113,97,129,101,137),(74,142,90,114,98,130,102,138),(75,143,81,115,99,121,103,139),(76,144,82,116,100,122,104,140),(77,145,83,117,91,123,105,131),(78,146,84,118,92,124,106,132),(79,147,85,119,93,125,107,133),(80,148,86,120,94,126,108,134)], [(1,93,52,79),(2,94,53,80),(3,95,54,71),(4,96,55,72),(5,97,56,73),(6,98,57,74),(7,99,58,75),(8,100,59,76),(9,91,60,77),(10,92,51,78),(11,124,153,146),(12,125,154,147),(13,126,155,148),(14,127,156,149),(15,128,157,150),(16,129,158,141),(17,130,159,142),(18,121,160,143),(19,122,151,144),(20,123,152,145),(21,117,40,131),(22,118,31,132),(23,119,32,133),(24,120,33,134),(25,111,34,135),(26,112,35,136),(27,113,36,137),(28,114,37,138),(29,115,38,139),(30,116,39,140),(41,86,63,108),(42,87,64,109),(43,88,65,110),(44,89,66,101),(45,90,67,102),(46,81,68,103),(47,82,69,104),(48,83,70,105),(49,84,61,106),(50,85,62,107)]])

70 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 20A ··· 20H 20I ··· 20X 40A ··· 40P order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4 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 Q16 C5×D4 C5×D4 C5×Q16 kernel C10×Q16 C2×C40 C5×Q16 Q8×C10 C2×Q16 C2×C8 Q16 C2×Q8 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×Q16 in GL4(𝔽41) generated by

 31 0 0 0 0 31 0 0 0 0 23 0 0 0 0 23
,
 33 38 0 0 8 8 0 0 0 0 29 12 0 0 29 29
,
 1 2 0 0 0 40 0 0 0 0 30 1 0 0 1 11
G:=sub<GL(4,GF(41))| [31,0,0,0,0,31,0,0,0,0,23,0,0,0,0,23],[33,8,0,0,38,8,0,0,0,0,29,29,0,0,12,29],[1,0,0,0,2,40,0,0,0,0,30,1,0,0,1,11] >;

C10×Q16 in GAP, Magma, Sage, TeX

C_{10}\times Q_{16}
% in TeX

G:=Group("C10xQ16");
// GroupNames label

G:=SmallGroup(160,195);
// by ID

G=gap.SmallGroup(160,195);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,487,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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