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## G = C4○D4×Dic5order 320 = 26·5

### Direct product of C4○D4 and Dic5

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4○D4×Dic5
G = < a,b,c,d,e | a4=c2=d10=1, b2=a2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 734 in 310 conjugacy classes, 195 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×10], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×15], C2×C4 [×20], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C42 [×10], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×6], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×Dic5 [×8], C2×Dic5 [×12], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C4×C4○D4, C4×Dic5, C4×Dic5 [×9], C4⋊Dic5 [×6], C23.D5 [×6], C22×Dic5 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4×Dic5 [×3], C23.21D10 [×3], D4×Dic5 [×6], Q8×Dic5 [×2], C10×C4○D4, C4○D4×Dic5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, Dic5 [×8], D10 [×7], C23×C4, C2×C4○D4 [×2], C2×Dic5 [×28], C22×D5 [×7], C4×C4○D4, C22×Dic5 [×14], C23×D5, D5×C4○D4 [×2], C23×Dic5, C4○D4×Dic5

Smallest permutation representation of C4○D4×Dic5
On 160 points
Generators in S160
(1 68 23 59)(2 69 24 60)(3 70 25 51)(4 61 26 52)(5 62 27 53)(6 63 28 54)(7 64 29 55)(8 65 30 56)(9 66 21 57)(10 67 22 58)(11 111 156 127)(12 112 157 128)(13 113 158 129)(14 114 159 130)(15 115 160 121)(16 116 151 122)(17 117 152 123)(18 118 153 124)(19 119 154 125)(20 120 155 126)(31 76 50 82)(32 77 41 83)(33 78 42 84)(34 79 43 85)(35 80 44 86)(36 71 45 87)(37 72 46 88)(38 73 47 89)(39 74 48 90)(40 75 49 81)(91 136 107 142)(92 137 108 143)(93 138 109 144)(94 139 110 145)(95 140 101 146)(96 131 102 147)(97 132 103 148)(98 133 104 149)(99 134 105 150)(100 135 106 141)
(1 54 23 63)(2 55 24 64)(3 56 25 65)(4 57 26 66)(5 58 27 67)(6 59 28 68)(7 60 29 69)(8 51 30 70)(9 52 21 61)(10 53 22 62)(11 116 156 122)(12 117 157 123)(13 118 158 124)(14 119 159 125)(15 120 160 126)(16 111 151 127)(17 112 152 128)(18 113 153 129)(19 114 154 130)(20 115 155 121)(31 71 50 87)(32 72 41 88)(33 73 42 89)(34 74 43 90)(35 75 44 81)(36 76 45 82)(37 77 46 83)(38 78 47 84)(39 79 48 85)(40 80 49 86)(91 147 107 131)(92 148 108 132)(93 149 109 133)(94 150 110 134)(95 141 101 135)(96 142 102 136)(97 143 103 137)(98 144 104 138)(99 145 105 139)(100 146 106 140)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 41)(10 42)(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 141)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 31)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)(21 149 26 144)(22 148 27 143)(23 147 28 142)(24 146 29 141)(25 145 30 150)(41 153 46 158)(42 152 47 157)(43 151 48 156)(44 160 49 155)(45 159 50 154)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 114 76 119)(72 113 77 118)(73 112 78 117)(74 111 79 116)(75 120 80 115)(81 126 86 121)(82 125 87 130)(83 124 88 129)(84 123 89 128)(85 122 90 127)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 4S ··· 4AD 5A 5B 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 D10 Dic5 D5×C4○D4 kernel C4○D4×Dic5 C2×C4×Dic5 C23.21D10 D4×Dic5 Q8×Dic5 C10×C4○D4 C5×C4○D4 C2×C4○D4 Dic5 C22×C4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 3 3 6 2 1 16 2 8 6 6 2 16 8

Matrix representation of C4○D4×Dic5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 32
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
,
 0 40 0 0 1 7 0 0 0 0 40 0 0 0 0 40
,
 23 13 0 0 16 18 0 0 0 0 9 0 0 0 0 9
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,40,7,0,0,0,0,40,0,0,0,0,40],[23,16,0,0,13,18,0,0,0,0,9,0,0,0,0,9] >;

C4○D4×Dic5 in GAP, Magma, Sage, TeX

C_4\circ D_4\times {\rm Dic}_5
% in TeX

G:=Group("C4oD4xDic5");
// GroupNames label

G:=SmallGroup(320,1498);
// by ID

G=gap.SmallGroup(320,1498);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,570,12550]);
// Polycyclic

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

׿
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𝔽