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## G = C4.Dic10order 160 = 25·5

### 3rd non-split extension by C4 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4.Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C4.Dic10
 Lower central C5 — C2×C10 — C4.Dic10
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4.Dic10
G = < a,b,c | a4=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 144 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C2×C4, C2×C4 [×2], C2×C4 [×4], C10 [×3], C42, C4⋊C4, C4⋊C4 [×5], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], C5×C4⋊C4, C4.Dic10
Quotients: C1, C2 [×7], C22 [×7], Q8 [×2], C23, D5, C2×Q8, C4○D4 [×2], D10 [×3], C42.C2, Dic10 [×2], C22×D5, C2×Dic10, D42D5, Q82D5, C4.Dic10

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 10A ··· 10F 20A ··· 20L 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 2 2 4 4 10 10 10 10 20 20 2 2 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C2 Q8 D5 C4○D4 D10 Dic10 D4⋊2D5 Q8⋊2D5 kernel C4.Dic10 C4×Dic5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C20 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 1 2 3 1 2 2 4 6 8 2 2

Matrix representation of C4.Dic10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 20 3 0 0 3 21
,
 11 2 0 0 25 27 0 0 0 0 0 1 0 0 40 0
,
 18 27 0 0 32 23 0 0 0 0 32 0 0 0 0 32
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,20,3,0,0,3,21],[11,25,0,0,2,27,0,0,0,0,0,40,0,0,1,0],[18,32,0,0,27,23,0,0,0,0,32,0,0,0,0,32] >;

C4.Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("C4.Dic10");
// GroupNames label

G:=SmallGroup(160,111);
// by ID

G=gap.SmallGroup(160,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,506,188,50,4613]);
// Polycyclic

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

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