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## G = C2×D5×C4⋊C4order 320 = 26·5

### Direct product of C2, D5 and C4⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×D5×C4⋊C4
G = < a,b,c,d,e | a2=b5=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1278 in 418 conjugacy classes, 207 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×12], C22, C22 [×6], C22 [×28], C5, C2×C4 [×10], C2×C4 [×50], C23, C23 [×14], D5 [×8], C10 [×3], C10 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×31], C24, Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×4], D10 [×28], C2×C10, C2×C10 [×6], C2×C4⋊C4, C2×C4⋊C4 [×11], C23×C4 [×3], C4×D5 [×16], C4×D5 [×16], C2×Dic5 [×10], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×D5 [×14], C22×C10, C22×C4⋊C4, C10.D4 [×8], C4⋊Dic5 [×4], C5×C4⋊C4 [×4], C2×C4×D5 [×20], C2×C4×D5 [×8], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, D5×C4⋊C4 [×8], C2×C10.D4 [×2], C2×C4⋊Dic5, C10×C4⋊C4, D5×C22×C4, D5×C22×C4 [×2], C2×D5×C4⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], D5, C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, D10 [×7], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C4×D5 [×4], C22×D5 [×7], C22×C4⋊C4, C2×C4×D5 [×6], D4×D5 [×2], Q8×D5 [×2], C23×D5, D5×C4⋊C4 [×4], D5×C22×C4, C2×D4×D5, C2×Q8×D5, C2×D5×C4⋊C4

Smallest permutation representation of C2×D5×C4⋊C4
On 160 points
Generators in S160
(1 89)(2 90)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)(122 125)(123 124)(127 130)(128 129)(132 135)(133 134)(137 140)(138 139)(142 145)(143 144)(147 150)(148 149)(152 155)(153 154)(157 160)(158 159)
(1 139 9 134)(2 140 10 135)(3 136 6 131)(4 137 7 132)(5 138 8 133)(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 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)(81 111 91 101)(82 112 92 102)(83 113 93 103)(84 114 94 104)(85 115 95 105)(86 116 96 106)(87 117 97 107)(88 118 98 108)(89 119 99 109)(90 120 100 110)(121 156 131 146)(122 157 132 147)(123 158 133 148)(124 159 134 149)(125 160 135 150)(126 151 136 141)(127 152 137 142)(128 153 138 143)(129 154 139 144)(130 155 140 145)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4L 4M ··· 4X 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 5 ··· 5 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 Q8 D5 D10 D10 C4×D5 D4×D5 Q8×D5 kernel C2×D5×C4⋊C4 D5×C4⋊C4 C2×C10.D4 C2×C4⋊Dic5 C10×C4⋊C4 D5×C22×C4 C2×C4×D5 C22×D5 C22×D5 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 2 1 1 3 16 4 4 2 8 6 16 4 4

Matrix representation of C2×D5×C4⋊C4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 34 0 0 0 0 0 0 7 1 0 0 0 0 33 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 23 27 0 0 0 0 32 18
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 37 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,32,0,0,0,0,27,18],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,37,40] >;

C2×D5×C4⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xD5xC4:C4");
// GroupNames label

G:=SmallGroup(320,1173);
// by ID

G=gap.SmallGroup(320,1173);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,80,12550]);
// Polycyclic

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

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