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## G = C4×D4⋊2D5order 320 = 26·5

### Direct product of C4 and D4⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×D4⋊2D5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C4×D4⋊2D5
 Lower central C5 — C10 — C4×D4⋊2D5
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C4×D42D5
G = < a,b,c,d,e | a4=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 862 in 310 conjugacy classes, 157 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×14], C22, C22 [×4], C22 [×8], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×31], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×4], C42, C42 [×9], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×8], Dic5 [×3], C20 [×4], C20 [×3], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C42 [×3], C42⋊C2 [×3], C4×D4, C4×D4 [×5], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×12], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5, C22×C10 [×2], C4×C4○D4, C4×Dic5 [×3], C4×Dic5 [×6], C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×3], D42D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, D5×C42, C23.11D10 [×2], Dic54D4 [×2], Dic53Q8, C4⋊C47D5, C2×C4×Dic5 [×2], C4×C5⋊D4 [×2], D4×Dic5, D4×C20, C2×D42D5, C4×D42D5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C4×C4○D4, C2×C4×D5 [×6], D42D5 [×2], C23×D5, D5×C22×C4, C2×D42D5, D5×C4○D4, C4×D42D5

Smallest permutation representation of C4×D42D5
On 160 points
Generators in S160
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)
(1 116 16 101)(2 117 17 102)(3 118 18 103)(4 119 19 104)(5 120 20 105)(6 111 11 106)(7 112 12 107)(8 113 13 108)(9 114 14 109)(10 115 15 110)(21 96 36 81)(22 97 37 82)(23 98 38 83)(24 99 39 84)(25 100 40 85)(26 91 31 86)(27 92 32 87)(28 93 33 88)(29 94 34 89)(30 95 35 90)(41 156 56 141)(42 157 57 142)(43 158 58 143)(44 159 59 144)(45 160 60 145)(46 151 51 146)(47 152 52 147)(48 153 53 148)(49 154 54 149)(50 155 55 150)(61 136 76 121)(62 137 77 122)(63 138 78 123)(64 139 79 124)(65 140 80 125)(66 131 71 126)(67 132 72 127)(68 133 73 128)(69 134 74 129)(70 135 75 130)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 25)(22 24)(26 30)(27 29)(31 35)(32 34)(36 40)(37 39)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)(81 85)(82 84)(86 90)(87 89)(91 95)(92 94)(96 100)(97 99)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 125)(122 124)(126 130)(127 129)(131 135)(132 134)(136 140)(137 139)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4L 4M ··· 4T 4U ··· 4AD 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 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 2 2 10 10 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 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4×D5 D4⋊2D5 D5×C4○D4 kernel C4×D4⋊2D5 C4×Dic10 D5×C42 C23.11D10 Dic5⋊4D4 Dic5⋊3Q8 C4⋊C4⋊7D5 C2×C4×Dic5 C4×C5⋊D4 D4×Dic5 D4×C20 C2×D4⋊2D5 D4⋊2D5 C4×D4 Dic5 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C4 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 16 2 4 4 2 4 2 4 2 16 4 4

Matrix representation of C4×D42D5 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 32 36 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 9 5 0 0 25 32
,
 0 40 0 0 1 34 0 0 0 0 1 0 0 0 0 1
,
 34 7 0 0 40 7 0 0 0 0 40 4 0 0 0 1
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,36,9],[1,0,0,0,0,1,0,0,0,0,9,25,0,0,5,32],[0,1,0,0,40,34,0,0,0,0,1,0,0,0,0,1],[34,40,0,0,7,7,0,0,0,0,40,0,0,0,4,1] >;

C4×D42D5 in GAP, Magma, Sage, TeX

C_4\times D_4\rtimes_2D_5
% in TeX

G:=Group("C4xD4:2D5");
// GroupNames label

G:=SmallGroup(320,1208);
// by ID

G=gap.SmallGroup(320,1208);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,794,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,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

׿
×
𝔽