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## G = C5×C42.29C22order 320 = 26·5

### Direct product of C5 and C42.29C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C42.29C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — D4×C10 — C5×D4⋊C4 — C5×C42.29C22
 Lower central C1 — C2 — C2×C4 — C5×C42.29C22
 Upper central C1 — C2×C10 — C4×C20 — C5×C42.29C22

Generators and relations for C5×C42.29C22
G = < a,b,c,d,e | a5=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 242 in 110 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C8⋊C4, D4⋊C4, C42.C2, C41D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C42.29C22, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, D4×C10, D4×C10, C5×C8⋊C4, C5×D4⋊C4, C5×C42.C2, C5×C41D4, C5×C42.29C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4.4D4, C8⋊C22, C5×D4, C22×C10, C42.29C22, D4×C10, C5×C4○D4, C5×C4.4D4, C5×C8⋊C22, C5×C42.29C22

Smallest permutation representation of C5×C42.29C22
On 160 points
Generators in S160
(1 37 109 29 101)(2 38 110 30 102)(3 39 111 31 103)(4 40 112 32 104)(5 33 105 25 97)(6 34 106 26 98)(7 35 107 27 99)(8 36 108 28 100)(9 44 116 22 93)(10 45 117 23 94)(11 46 118 24 95)(12 47 119 17 96)(13 48 120 18 89)(14 41 113 19 90)(15 42 114 20 91)(16 43 115 21 92)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 84 153 73 145)(66 85 154 74 146)(67 86 155 75 147)(68 87 156 76 148)(69 88 157 77 149)(70 81 158 78 150)(71 82 159 79 151)(72 83 160 80 152)
(1 68 125 116)(2 65 126 113)(3 70 127 118)(4 67 128 115)(5 72 121 120)(6 69 122 117)(7 66 123 114)(8 71 124 119)(9 29 76 133)(10 26 77 130)(11 31 78 135)(12 28 79 132)(13 25 80 129)(14 30 73 134)(15 27 74 131)(16 32 75 136)(17 36 82 140)(18 33 83 137)(19 38 84 142)(20 35 85 139)(21 40 86 144)(22 37 87 141)(23 34 88 138)(24 39 81 143)(41 102 145 54)(42 99 146 51)(43 104 147 56)(44 101 148 53)(45 98 149 50)(46 103 150 55)(47 100 151 52)(48 97 152 49)(57 89 105 160)(58 94 106 157)(59 91 107 154)(60 96 108 159)(61 93 109 156)(62 90 110 153)(63 95 111 158)(64 92 112 155)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(1 116)(2 71)(3 114)(4 69)(5 120)(6 67)(7 118)(8 65)(9 29)(10 136)(11 27)(12 134)(13 25)(14 132)(15 31)(16 130)(17 142)(18 33)(19 140)(20 39)(21 138)(22 37)(23 144)(24 35)(26 75)(28 73)(30 79)(32 77)(34 86)(36 84)(38 82)(40 88)(41 52)(42 103)(43 50)(44 101)(45 56)(46 99)(47 54)(48 97)(49 152)(51 150)(53 148)(55 146)(57 160)(58 92)(59 158)(60 90)(61 156)(62 96)(63 154)(64 94)(66 127)(68 125)(70 123)(72 121)(74 135)(76 133)(78 131)(80 129)(81 139)(83 137)(85 143)(87 141)(89 105)(91 111)(93 109)(95 107)(98 147)(100 145)(102 151)(104 149)(106 155)(108 153)(110 159)(112 157)(113 124)(115 122)(117 128)(119 126)
(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)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20H 20I ··· 20P 20Q ··· 20X 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 8 8 2 2 4 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 8 ··· 8 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 D4 C4○D4 C5×D4 C5×C4○D4 C8⋊C22 C5×C8⋊C22 kernel C5×C42.29C22 C5×C8⋊C4 C5×D4⋊C4 C5×C42.C2 C5×C4⋊1D4 C42.29C22 C8⋊C4 D4⋊C4 C42.C2 C4⋊1D4 C2×C20 C20 C2×C4 C4 C10 C2 # reps 1 1 4 1 1 4 4 16 4 4 2 4 8 16 2 8

Matrix representation of C5×C42.29C22 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 18 23 0 0 0 0 34 23 0 0 0 0 0 0 29 0 20 21 0 0 0 29 20 20 0 0 20 20 12 0 0 0 21 20 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 18 23 0 0 0 0 2 23 0 0 0 0 0 0 29 0 20 21 0 0 0 12 21 21 0 0 20 21 0 29 0 0 21 21 29 0
,
 2 39 0 0 0 0 22 39 0 0 0 0 0 0 21 20 0 12 0 0 21 21 29 0 0 0 29 0 20 21 0 0 0 29 20 20

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[18,34,0,0,0,0,23,23,0,0,0,0,0,0,29,0,20,21,0,0,0,29,20,20,0,0,20,20,12,0,0,0,21,20,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[18,2,0,0,0,0,23,23,0,0,0,0,0,0,29,0,20,21,0,0,0,12,21,21,0,0,20,21,0,29,0,0,21,21,29,0],[2,22,0,0,0,0,39,39,0,0,0,0,0,0,21,21,29,0,0,0,20,21,0,29,0,0,0,29,20,20,0,0,12,0,21,20] >;

C5×C42.29C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2._{29}C_2^2
% in TeX

G:=Group("C5xC4^2.29C2^2");
// GroupNames label

G:=SmallGroup(320,991);
// by ID

G=gap.SmallGroup(320,991);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1688,1766,1731,226,7004,172,10085,124]);
// Polycyclic

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

׿
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