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## G = C2×C10.D4order 160 = 25·5

### Direct product of C2 and C10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C10.D4
 Lower central C5 — C10 — C2×C10.D4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×C10.D4
G = < a,b,c,d | a2=b10=c4=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 216 in 92 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C22, C22 [×6], C5, C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×C4⋊C4, C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C10.D4 [×4], C22×Dic5 [×2], C22×C20, C2×C10.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4

Smallest permutation representation of C2×C10.D4
Regular action on 160 points
Generators in S160
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 119)(12 120)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 90)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(91 146)(92 147)(93 148)(94 149)(95 150)(96 141)(97 142)(98 143)(99 144)(100 145)(101 136)(102 137)(103 138)(104 139)(105 140)(106 131)(107 132)(108 133)(109 134)(110 135)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)
(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)
(1 110 25 100)(2 109 26 99)(3 108 27 98)(4 107 28 97)(5 106 29 96)(6 105 30 95)(7 104 21 94)(8 103 22 93)(9 102 23 92)(10 101 24 91)(11 90 154 76)(12 89 155 75)(13 88 156 74)(14 87 157 73)(15 86 158 72)(16 85 159 71)(17 84 160 80)(18 83 151 79)(19 82 152 78)(20 81 153 77)(31 129 46 119)(32 128 47 118)(33 127 48 117)(34 126 49 116)(35 125 50 115)(36 124 41 114)(37 123 42 113)(38 122 43 112)(39 121 44 111)(40 130 45 120)(51 149 66 139)(52 148 67 138)(53 147 68 137)(54 146 69 136)(55 145 70 135)(56 144 61 134)(57 143 62 133)(58 142 63 132)(59 141 64 131)(60 150 65 140)
(1 115 6 120)(2 114 7 119)(3 113 8 118)(4 112 9 117)(5 111 10 116)(11 61 16 66)(12 70 17 65)(13 69 18 64)(14 68 19 63)(15 67 20 62)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 143 86 148)(82 142 87 147)(83 141 88 146)(84 150 89 145)(85 149 90 144)

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

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

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

52 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 20A ··· 20P order 1 2 ··· 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + + + - image C1 C2 C2 C2 C4 D4 Q8 D5 D10 D10 Dic10 C4×D5 C5⋊D4 kernel C2×C10.D4 C10.D4 C22×Dic5 C22×C20 C2×Dic5 C2×C10 C2×C10 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 8 2 2 2 4 2 8 8 8

Matrix representation of C2×C10.D4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 1 0 0 0 0 34 6 0 0 34 0
,
 1 0 0 0 0 1 0 0 0 0 24 23 0 0 7 17
,
 9 0 0 0 0 1 0 0 0 0 1 40 0 0 0 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,34,34,0,0,6,0],[1,0,0,0,0,1,0,0,0,0,24,7,0,0,23,17],[9,0,0,0,0,1,0,0,0,0,1,0,0,0,40,40] >;

C2×C10.D4 in GAP, Magma, Sage, TeX

C_2\times C_{10}.D_4
% in TeX

G:=Group("C2xC10.D4");
// GroupNames label

G:=SmallGroup(160,144);
// by ID

G=gap.SmallGroup(160,144);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,362,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^4=1,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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