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## G = C2×C4⋊Dic5order 160 = 25·5

### Direct product of C2 and C4⋊Dic5

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4⋊Dic5
G = < a,b,c,d | a2=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 216 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×8], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C4⋊Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4⋊Dic5
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, Dic5 [×4], D10 [×3], C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic5

Smallest permutation representation of C2×C4⋊Dic5
Regular action on 160 points
Generators in S160
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 51)(30 52)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 81)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 141)(99 142)(100 143)(101 138)(102 139)(103 140)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(121 158)(122 159)(123 160)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)
(1 47 27 34)(2 48 28 35)(3 49 29 36)(4 50 30 37)(5 41 21 38)(6 42 22 39)(7 43 23 40)(8 44 24 31)(9 45 25 32)(10 46 26 33)(11 140 160 148)(12 131 151 149)(13 132 152 150)(14 133 153 141)(15 134 154 142)(16 135 155 143)(17 136 156 144)(18 137 157 145)(19 138 158 146)(20 139 159 147)(51 87 64 71)(52 88 65 72)(53 89 66 73)(54 90 67 74)(55 81 68 75)(56 82 69 76)(57 83 70 77)(58 84 61 78)(59 85 62 79)(60 86 63 80)(91 111 109 129)(92 112 110 130)(93 113 101 121)(94 114 102 122)(95 115 103 123)(96 116 104 124)(97 117 105 125)(98 118 106 126)(99 119 107 127)(100 120 108 128)
(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 95 6 100)(2 94 7 99)(3 93 8 98)(4 92 9 97)(5 91 10 96)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 118 36 113)(32 117 37 112)(33 116 38 111)(34 115 39 120)(35 114 40 119)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 138 56 133)(52 137 57 132)(53 136 58 131)(54 135 59 140)(55 134 60 139)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 158 76 153)(72 157 77 152)(73 156 78 151)(74 155 79 160)(75 154 80 159)

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

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

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

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 Dic5 D10 D10 Dic10 D20 kernel C2×C4⋊Dic5 C4⋊Dic5 C22×Dic5 C22×C20 C2×C20 C2×C10 C2×C10 C22×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 8 2 2 2 8 4 2 8 8

Matrix representation of C2×C4⋊Dic5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 1 9
,
 35 1 0 0 0 0 40 0 0 0 0 0 0 0 35 1 0 0 0 0 40 0 0 0 0 0 0 0 16 0 0 0 0 0 32 18
,
 14 2 0 0 0 0 4 27 0 0 0 0 0 0 27 39 0 0 0 0 37 14 0 0 0 0 0 0 25 40 0 0 0 0 9 16

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,1,0,0,0,0,0,9],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,35,40,0,0,0,0,1,0,0,0,0,0,0,0,16,32,0,0,0,0,0,18],[14,4,0,0,0,0,2,27,0,0,0,0,0,0,27,37,0,0,0,0,39,14,0,0,0,0,0,0,25,9,0,0,0,0,40,16] >;

C2×C4⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xC4:Dic5");
// GroupNames label

G:=SmallGroup(160,146);
// by ID

G=gap.SmallGroup(160,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,4613]);
// Polycyclic

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

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