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## G = C4×Dic10order 160 = 25·5

### Direct product of C4 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C4×Dic10
 Lower central C5 — C10 — C4×Dic10
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×Dic10
G = < a,b,c | a4=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 168 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C4 [×4], C4 [×7], C22, C5, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42, C42 [×2], C4⋊C4 [×3], C2×Q8, Dic5 [×4], Dic5 [×2], C20 [×4], C20, C2×C10, C4×Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20 [×3], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, C22×C4, C2×Q8, C4○D4, D10 [×3], C4×Q8, Dic10 [×2], C4×D5 [×2], C22×D5, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10

Smallest permutation representation of C4×Dic10
Regular action on 160 points
Generators in S160
(1 59 79 120)(2 60 80 101)(3 41 61 102)(4 42 62 103)(5 43 63 104)(6 44 64 105)(7 45 65 106)(8 46 66 107)(9 47 67 108)(10 48 68 109)(11 49 69 110)(12 50 70 111)(13 51 71 112)(14 52 72 113)(15 53 73 114)(16 54 74 115)(17 55 75 116)(18 56 76 117)(19 57 77 118)(20 58 78 119)(21 127 89 151)(22 128 90 152)(23 129 91 153)(24 130 92 154)(25 131 93 155)(26 132 94 156)(27 133 95 157)(28 134 96 158)(29 135 97 159)(30 136 98 160)(31 137 99 141)(32 138 100 142)(33 139 81 143)(34 140 82 144)(35 121 83 145)(36 122 84 146)(37 123 85 147)(38 124 86 148)(39 125 87 149)(40 126 88 150)
(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)
(1 141 11 151)(2 160 12 150)(3 159 13 149)(4 158 14 148)(5 157 15 147)(6 156 16 146)(7 155 17 145)(8 154 18 144)(9 153 19 143)(10 152 20 142)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(61 135 71 125)(62 134 72 124)(63 133 73 123)(64 132 74 122)(65 131 75 121)(66 130 76 140)(67 129 77 139)(68 128 78 138)(69 127 79 137)(70 126 80 136)(81 108 91 118)(82 107 92 117)(83 106 93 116)(84 105 94 115)(85 104 95 114)(86 103 96 113)(87 102 97 112)(88 101 98 111)(89 120 99 110)(90 119 100 109)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 10A ··· 10F 20A ··· 20X order 1 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 1 1 1 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 Q8 D5 C4○D4 D10 Dic10 C4×D5 C4○D20 kernel C4×Dic10 C4×Dic5 C10.D4 C4⋊Dic5 C4×C20 C2×Dic10 Dic10 C20 C42 C10 C2×C4 C4 C4 C2 # reps 1 2 2 1 1 1 8 2 2 2 6 8 8 8

Matrix representation of C4×Dic10 in GL3(𝔽41) generated by

 32 0 0 0 9 0 0 0 9
,
 40 0 0 0 9 11 0 30 14
,
 40 0 0 0 27 12 0 28 14
G:=sub<GL(3,GF(41))| [32,0,0,0,9,0,0,0,9],[40,0,0,0,9,30,0,11,14],[40,0,0,0,27,28,0,12,14] >;

C4×Dic10 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{10}
% in TeX

G:=Group("C4xDic10");
// GroupNames label

G:=SmallGroup(160,89);
// by ID

G=gap.SmallGroup(160,89);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,50,4613]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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