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## G = C2×Dic5.5D4order 320 = 26·5

### Direct product of C2 and Dic5.5D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×Dic5.5D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D10⋊C4 — C2×Dic5.5D4
 Lower central C5 — C2×C10 — C2×Dic5.5D4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×Dic5.5D4
G = < a,b,c,d,e | a2=b10=d4=1, c2=e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=b5d-1 >

Subgroups: 1214 in 330 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C22, C22 [×6], C22 [×20], C5, C2×C4 [×4], C2×C4 [×18], D4 [×8], Q8 [×8], C23, C23 [×2], C23 [×14], D5 [×2], C10 [×3], C10 [×4], C10 [×2], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×12], C22×C4 [×2], C22×C4 [×3], C2×D4 [×8], C2×Q8 [×8], C24, C24, Dic5 [×4], Dic5 [×4], C20 [×4], D10 [×10], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×3], C4.4D4 [×8], C22×D4, C22×Q8, Dic10 [×8], C2×Dic5 [×10], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×4], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C4.4D4, C4×Dic5 [×4], D10⋊C4 [×8], C23.D5 [×4], C5×C22⋊C4 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×3], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, Dic5.5D4 [×8], C2×C4×Dic5, C2×D10⋊C4 [×2], C2×C23.D5, C10×C22⋊C4, C22×Dic10, C22×C5⋊D4, C2×Dic5.5D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C2×C4.4D4, C4○D20 [×2], D4×D5 [×2], D42D5 [×2], C23×D5, Dic5.5D4 [×4], C2×C4○D20, C2×D4×D5, C2×D42D5, C2×Dic5.5D4

Smallest permutation representation of C2×Dic5.5D4
On 160 points
Generators in S160
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(11 79)(12 80)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 101)(28 102)(29 103)(30 104)(31 127)(32 128)(33 129)(34 130)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 111)(48 112)(49 113)(50 114)(51 147)(52 148)(53 149)(54 150)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 131)(68 132)(69 133)(70 134)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 151)(88 152)(89 153)(90 154)
(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)
(1 116 6 111)(2 115 7 120)(3 114 8 119)(4 113 9 118)(5 112 10 117)(11 58 16 53)(12 57 17 52)(13 56 18 51)(14 55 19 60)(15 54 20 59)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 102 36 107)(32 101 37 106)(33 110 38 105)(34 109 39 104)(35 108 40 103)(41 92 46 97)(42 91 47 96)(43 100 48 95)(44 99 49 94)(45 98 50 93)(61 157 66 152)(62 156 67 151)(63 155 68 160)(64 154 69 159)(65 153 70 158)(71 142 76 147)(72 141 77 146)(73 150 78 145)(74 149 79 144)(75 148 80 143)(81 132 86 137)(82 131 87 136)(83 140 88 135)(84 139 89 134)(85 138 90 133)
(1 56 23 67)(2 57 24 68)(3 58 25 69)(4 59 26 70)(5 60 27 61)(6 51 28 62)(7 52 29 63)(8 53 30 64)(9 54 21 65)(10 55 22 66)(11 130 154 119)(12 121 155 120)(13 122 156 111)(14 123 157 112)(15 124 158 113)(16 125 159 114)(17 126 160 115)(18 127 151 116)(19 128 152 117)(20 129 153 118)(31 87 42 76)(32 88 43 77)(33 89 44 78)(34 90 45 79)(35 81 46 80)(36 82 47 71)(37 83 48 72)(38 84 49 73)(39 85 50 74)(40 86 41 75)(91 147 102 136)(92 148 103 137)(93 149 104 138)(94 150 105 139)(95 141 106 140)(96 142 107 131)(97 143 108 132)(98 144 109 133)(99 145 110 134)(100 146 101 135)
(1 51 6 56)(2 60 7 55)(3 59 8 54)(4 58 9 53)(5 57 10 52)(11 118 16 113)(12 117 17 112)(13 116 18 111)(14 115 19 120)(15 114 20 119)(21 64 26 69)(22 63 27 68)(23 62 28 67)(24 61 29 66)(25 70 30 65)(31 87 36 82)(32 86 37 81)(33 85 38 90)(34 84 39 89)(35 83 40 88)(41 77 46 72)(42 76 47 71)(43 75 48 80)(44 74 49 79)(45 73 50 78)(91 142 96 147)(92 141 97 146)(93 150 98 145)(94 149 99 144)(95 148 100 143)(101 132 106 137)(102 131 107 136)(103 140 108 135)(104 139 109 134)(105 138 110 133)(121 157 126 152)(122 156 127 151)(123 155 128 160)(124 154 129 159)(125 153 130 158)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 20 20 2 2 2 2 4 4 10 ··· 10 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C4○D20 D4×D5 D4⋊2D5 kernel C2×Dic5.5D4 Dic5.5D4 C2×C4×Dic5 C2×D10⋊C4 C2×C23.D5 C10×C22⋊C4 C22×Dic10 C22×C5⋊D4 C2×Dic5 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C22 C22 C22 # reps 1 8 1 2 1 1 1 1 4 2 8 8 4 2 16 4 4

Matrix representation of C2×Dic5.5D4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 7 1 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 38 17 0 0 0 38 3 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 9 39 0 0 0 0 32
,
 40 0 0 0 0 0 19 32 0 0 0 22 22 0 0 0 0 0 32 2 0 0 0 1 9

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,7,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,38,38,0,0,0,17,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,39,32],[40,0,0,0,0,0,19,22,0,0,0,32,22,0,0,0,0,0,32,1,0,0,0,2,9] >;

C2×Dic5.5D4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5._5D_4
% in TeX

G:=Group("C2xDic5.5D4");
// GroupNames label

G:=SmallGroup(320,1163);
// by ID

G=gap.SmallGroup(320,1163);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,1571,297,12550]);
// Polycyclic

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

׿
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