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G = C5×D4.2D4order 320 = 26·5

Direct product of C5 and D4.2D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.2D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — D4×C10 — C5×C4.4D4 — C5×D4.2D4
 Lower central C1 — C2 — C2×C4 — C5×D4.2D4
 Upper central C1 — C2×C10 — C4×C20 — C5×D4.2D4

Generators and relations for C5×D4.2D4
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 250 in 124 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D4.2D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C5×SD16, C22×C20, D4×C10, Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C4⋊C8, D4×C20, C5×C4.4D4, C10×D8, C10×SD16, C5×D4.2D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C4○D8, C8⋊C22, C5×D4, C22×C10, D4.2D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C4○D8, C5×C8⋊C22, C5×D4.2D4

Smallest permutation representation of C5×D4.2D4
On 160 points
Generators in S160
(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 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 23 144 156)(7 24 145 157)(8 25 141 158)(9 21 142 159)(10 22 143 160)(16 33 137 154)(17 34 138 155)(18 35 139 151)(19 31 140 152)(20 32 136 153)(26 43 50 52)(27 44 46 53)(28 45 47 54)(29 41 48 55)(30 42 49 51)(56 73 94 87)(57 74 95 88)(58 75 91 89)(59 71 92 90)(60 72 93 86)(61 78 82 107)(62 79 83 108)(63 80 84 109)(64 76 85 110)(65 77 81 106)(96 127 134 113)(97 128 135 114)(98 129 131 115)(99 130 132 111)(100 126 133 112)(101 147 122 118)(102 148 123 119)(103 149 124 120)(104 150 125 116)(105 146 121 117)
(1 77)(2 78)(3 79)(4 80)(5 76)(6 150)(7 146)(8 147)(9 148)(10 149)(11 110)(12 106)(13 107)(14 108)(15 109)(16 128)(17 129)(18 130)(19 126)(20 127)(21 102)(22 103)(23 104)(24 105)(25 101)(26 59)(27 60)(28 56)(29 57)(30 58)(31 100)(32 96)(33 97)(34 98)(35 99)(36 64)(37 65)(38 61)(39 62)(40 63)(41 88)(42 89)(43 90)(44 86)(45 87)(46 93)(47 94)(48 95)(49 91)(50 92)(51 75)(52 71)(53 72)(54 73)(55 74)(66 81)(67 82)(68 83)(69 84)(70 85)(111 139)(112 140)(113 136)(114 137)(115 138)(116 144)(117 145)(118 141)(119 142)(120 143)(121 157)(122 158)(123 159)(124 160)(125 156)(131 155)(132 151)(133 152)(134 153)(135 154)
(1 134 54 121)(2 135 55 122)(3 131 51 123)(4 132 52 124)(5 133 53 125)(6 85 19 93)(7 81 20 94)(8 82 16 95)(9 83 17 91)(10 84 18 92)(11 100 44 104)(12 96 45 105)(13 97 41 101)(14 98 42 102)(15 99 43 103)(21 108 34 89)(22 109 35 90)(23 110 31 86)(24 106 32 87)(25 107 33 88)(26 120 40 111)(27 116 36 112)(28 117 37 113)(29 118 38 114)(30 119 39 115)(46 150 70 126)(47 146 66 127)(48 147 67 128)(49 148 68 129)(50 149 69 130)(56 145 65 136)(57 141 61 137)(58 142 62 138)(59 143 63 139)(60 144 64 140)(71 160 80 151)(72 156 76 152)(73 157 77 153)(74 158 78 154)(75 159 79 155)
(1 105 12 121)(2 101 13 122)(3 102 14 123)(4 103 15 124)(5 104 11 125)(6 110 144 76)(7 106 145 77)(8 107 141 78)(9 108 142 79)(10 109 143 80)(16 88 137 74)(17 89 138 75)(18 90 139 71)(19 86 140 72)(20 87 136 73)(21 83 159 62)(22 84 160 63)(23 85 156 64)(24 81 157 65)(25 82 158 61)(26 111 50 130)(27 112 46 126)(28 113 47 127)(29 114 48 128)(30 115 49 129)(31 93 152 60)(32 94 153 56)(33 95 154 57)(34 91 155 58)(35 92 151 59)(36 116 70 150)(37 117 66 146)(38 118 67 147)(39 119 68 148)(40 120 69 149)(41 135 55 97)(42 131 51 98)(43 132 52 99)(44 133 53 100)(45 134 54 96)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 20A ··· 20P 20Q ··· 20AB 20AC 20AD 20AE 20AF 40A ··· 40P order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 8 2 2 2 2 4 4 4 8 1 1 1 1 4 4 4 4 1 ··· 1 4 ··· 4 8 8 8 8 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 D4 D4 C4○D4 C4○D8 C5×D4 C5×D4 C5×C4○D4 C5×C4○D8 C8⋊C22 C5×C8⋊C22 kernel C5×D4.2D4 C5×D4⋊C4 C5×Q8⋊C4 C5×C4⋊C8 D4×C20 C5×C4.4D4 C10×D8 C10×SD16 D4.2D4 D4⋊C4 Q8⋊C4 C4⋊C8 C4×D4 C4.4D4 C2×D8 C2×SD16 C2×C20 C5×D4 C20 C10 C2×C4 D4 C4 C2 C10 C2 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 2 2 2 4 8 8 8 16 1 4

Matrix representation of C5×D4.2D4 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 0 40 0 0 40 0 0 0 0 0 12 29 0 0 29 29
,
 0 9 0 0 9 0 0 0 0 0 32 0 0 0 0 32
,
 0 32 0 0 9 0 0 0 0 0 32 0 0 0 0 9
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,40,0,0,0,0,0,12,29,0,0,29,29],[0,9,0,0,9,0,0,0,0,0,32,0,0,0,0,32],[0,9,0,0,32,0,0,0,0,0,32,0,0,0,0,9] >;

C5×D4.2D4 in GAP, Magma, Sage, TeX

C_5\times D_4._2D_4
% in TeX

G:=Group("C5xD4.2D4");
// GroupNames label

G:=SmallGroup(320,964);
// by ID

G=gap.SmallGroup(320,964);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽