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G = C2×C42⋊D5order 320 = 26·5

Direct product of C2 and C42⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C42⋊D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×C42⋊D5
 Lower central C5 — C10 — C2×C42⋊D5
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C42⋊D5
G = < a,b,c,d,e | a2=b4=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 990 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×6], C22 [×16], C5, C2×C4 [×10], C2×C4 [×34], C23, C23 [×10], D5 [×4], C10, C10 [×6], C42 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×15], C24, Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C4×D5 [×16], C2×Dic5 [×10], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C42⋊C2, C4×Dic5 [×4], C10.D4 [×8], D10⋊C4 [×8], C4×C20 [×4], C2×C4×D5 [×12], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C42⋊D5 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×D10⋊C4 [×2], C2×C4×C20, D5×C22×C4, C2×C42⋊D5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C2×C42⋊C2, C2×C4×D5 [×6], C4○D20 [×4], C23×D5, C42⋊D5 [×4], D5×C22×C4, C2×C4○D20 [×2], C2×C42⋊D5

Smallest permutation representation of C2×C42⋊D5
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 86)(12 87)(13 88)(14 89)(15 90)(16 81)(17 82)(18 83)(19 84)(20 85)(21 116)(22 117)(23 118)(24 119)(25 120)(26 111)(27 112)(28 113)(29 114)(30 115)(31 106)(32 107)(33 108)(34 109)(35 110)(36 101)(37 102)(38 103)(39 104)(40 105)(41 136)(42 137)(43 138)(44 139)(45 140)(46 131)(47 132)(48 133)(49 134)(50 135)(51 126)(52 127)(53 128)(54 129)(55 130)(56 121)(57 122)(58 123)(59 124)(60 125)(61 156)(62 157)(63 158)(64 159)(65 160)(66 151)(67 152)(68 153)(69 154)(70 155)(71 146)(72 147)(73 148)(74 149)(75 150)(76 141)(77 142)(78 143)(79 144)(80 145)
(1 136 6 131)(2 137 7 132)(3 138 8 133)(4 139 9 134)(5 140 10 135)(11 126 16 121)(12 127 17 122)(13 128 18 123)(14 129 19 124)(15 130 20 125)(21 156 26 151)(22 157 27 152)(23 158 28 153)(24 159 29 154)(25 160 30 155)(31 146 36 141)(32 147 37 142)(33 148 38 143)(34 149 39 144)(35 150 40 145)(41 91 46 96)(42 92 47 97)(43 93 48 98)(44 94 49 99)(45 95 50 100)(51 81 56 86)(52 82 57 87)(53 83 58 88)(54 84 59 89)(55 85 60 90)(61 111 66 116)(62 112 67 117)(63 113 68 118)(64 114 69 119)(65 115 70 120)(71 101 76 106)(72 102 77 107)(73 103 78 108)(74 104 79 109)(75 105 80 110)
(1 76 16 61)(2 77 17 62)(3 78 18 63)(4 79 19 64)(5 80 20 65)(6 71 11 66)(7 72 12 67)(8 73 13 68)(9 74 14 69)(10 75 15 70)(21 46 36 51)(22 47 37 52)(23 48 38 53)(24 49 39 54)(25 50 40 55)(26 41 31 56)(27 42 32 57)(28 43 33 58)(29 44 34 59)(30 45 35 60)(81 156 96 141)(82 157 97 142)(83 158 98 143)(84 159 99 144)(85 160 100 145)(86 151 91 146)(87 152 92 147)(88 153 93 148)(89 154 94 149)(90 155 95 150)(101 126 116 131)(102 127 117 132)(103 128 118 133)(104 129 119 134)(105 130 120 135)(106 121 111 136)(107 122 112 137)(108 123 113 138)(109 124 114 139)(110 125 115 140)
(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)(2 94)(3 93)(4 92)(5 91)(6 100)(7 99)(8 98)(9 97)(10 96)(11 85)(12 84)(13 83)(14 82)(15 81)(16 90)(17 89)(18 88)(19 87)(20 86)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 135)(42 134)(43 133)(44 132)(45 131)(46 140)(47 139)(48 138)(49 137)(50 136)(51 125)(52 124)(53 123)(54 122)(55 121)(56 130)(57 129)(58 128)(59 127)(60 126)(61 160)(62 159)(63 158)(64 157)(65 156)(66 155)(67 154)(68 153)(69 152)(70 151)(71 150)(72 149)(73 148)(74 147)(75 146)(76 145)(77 144)(78 143)(79 142)(80 141)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 C4×D5 C4○D20 kernel C2×C42⋊D5 C42⋊D5 C2×C4×Dic5 C2×C10.D4 C2×D10⋊C4 C2×C4×C20 D5×C22×C4 C2×C4×D5 C2×C42 C2×C10 C42 C22×C4 C2×C4 C22 # reps 1 8 1 2 2 1 1 16 2 8 8 6 16 32

Matrix representation of C2×C42⋊D5 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 32 0 0 0 0 0 32
,
 40 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 39 28 0 0 0 13 2
,
 1 0 0 0 0 0 40 40 0 0 0 36 35 0 0 0 0 0 6 40 0 0 0 1 0
,
 1 0 0 0 0 0 0 7 0 0 0 6 0 0 0 0 0 0 35 1 0 0 0 6 6

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,39,13,0,0,0,28,2],[1,0,0,0,0,0,40,36,0,0,0,40,35,0,0,0,0,0,6,1,0,0,0,40,0],[1,0,0,0,0,0,0,6,0,0,0,7,0,0,0,0,0,0,35,6,0,0,0,1,6] >;

C2×C42⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes D_5
% in TeX

G:=Group("C2xC4^2:D5");
// GroupNames label

G:=SmallGroup(320,1144);
// by ID

G=gap.SmallGroup(320,1144);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,1123,80,12550]);
// Polycyclic

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

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