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## G = C5×D4×Q8order 320 = 26·5

### Direct product of C5, D4 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×D4×Q8
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — Q8×C10 — C5×C22⋊Q8 — C5×D4×Q8
 Lower central C1 — C22 — C5×D4×Q8
 Upper central C1 — C2×C10 — C5×D4×Q8

Generators and relations for C5×D4×Q8
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 378 in 280 conjugacy classes, 182 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×9], C22, C22 [×4], C22 [×4], C5, C2×C4, C2×C4 [×12], C2×C4 [×12], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×8], C20 [×8], C20 [×9], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], C2×C20, C2×C20 [×12], C2×C20 [×12], C5×D4 [×4], C5×Q8 [×4], C5×Q8 [×12], C22×C10 [×2], D4×Q8, C4×C20 [×3], C5×C22⋊C4 [×6], C5×C4⋊C4 [×12], C22×C20 [×6], D4×C10, Q8×C10, Q8×C10 [×6], Q8×C10 [×8], D4×C20 [×3], Q8×C20, C5×C22⋊Q8 [×6], C5×C4⋊Q8 [×3], Q8×C2×C10 [×2], C5×D4×Q8
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C2×C10 [×35], C22×D4, C22×Q8, 2- 1+4, C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], D4×Q8, D4×C10 [×6], Q8×C10 [×6], C23×C10, D4×C2×C10, Q8×C2×C10, C5×2- 1+4, C5×D4×Q8

Smallest permutation representation of C5×D4×Q8
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)
(1 123 30 135)(2 124 26 131)(3 125 27 132)(4 121 28 133)(5 122 29 134)(6 70 156 56)(7 66 157 57)(8 67 158 58)(9 68 159 59)(10 69 160 60)(11 75 16 63)(12 71 17 64)(13 72 18 65)(14 73 19 61)(15 74 20 62)(21 119 33 128)(22 120 34 129)(23 116 35 130)(24 117 31 126)(25 118 32 127)(36 110 50 96)(37 106 46 97)(38 107 47 98)(39 108 48 99)(40 109 49 100)(41 113 53 101)(42 114 54 102)(43 115 55 103)(44 111 51 104)(45 112 52 105)(76 143 90 155)(77 144 86 151)(78 145 87 152)(79 141 88 153)(80 142 89 154)(81 139 93 148)(82 140 94 149)(83 136 95 150)(84 137 91 146)(85 138 92 147)
(1 135)(2 131)(3 132)(4 133)(5 134)(6 70)(7 66)(8 67)(9 68)(10 69)(11 75)(12 71)(13 72)(14 73)(15 74)(16 63)(17 64)(18 65)(19 61)(20 62)(21 128)(22 129)(23 130)(24 126)(25 127)(26 124)(27 125)(28 121)(29 122)(30 123)(31 117)(32 118)(33 119)(34 120)(35 116)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 156)(57 157)(58 158)(59 159)(60 160)(76 155)(77 151)(78 152)(79 153)(80 154)(81 148)(82 149)(83 150)(84 146)(85 147)(86 144)(87 145)(88 141)(89 142)(90 143)(91 137)(92 138)(93 139)(94 140)(95 136)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 143 16 150)(7 144 17 146)(8 145 18 147)(9 141 19 148)(10 142 20 149)(11 136 156 155)(12 137 157 151)(13 138 158 152)(14 139 159 153)(15 140 160 154)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)(56 76 75 95)(57 77 71 91)(58 78 72 92)(59 79 73 93)(60 80 74 94)(61 81 68 88)(62 82 69 89)(63 83 70 90)(64 84 66 86)(65 85 67 87)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 128 108 121)(102 129 109 122)(103 130 110 123)(104 126 106 124)(105 127 107 125)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 123 16 130)(7 124 17 126)(8 125 18 127)(9 121 19 128)(10 122 20 129)(11 116 156 135)(12 117 157 131)(13 118 158 132)(14 119 159 133)(15 120 160 134)(21 68 28 61)(22 69 29 62)(23 70 30 63)(24 66 26 64)(25 67 27 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 88 48 81)(42 89 49 82)(43 90 50 83)(44 86 46 84)(45 87 47 85)(96 136 115 155)(97 137 111 151)(98 138 112 152)(99 139 113 153)(100 140 114 154)(101 141 108 148)(102 142 109 149)(103 143 110 150)(104 144 106 146)(105 145 107 147)

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

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

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

125 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 4I ··· 4Q 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AB 20A ··· 20AF 20AG ··· 20BP order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

125 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 Q8 D4 C5×Q8 C5×D4 2- 1+4 C5×2- 1+4 kernel C5×D4×Q8 D4×C20 Q8×C20 C5×C22⋊Q8 C5×C4⋊Q8 Q8×C2×C10 D4×Q8 C4×D4 C4×Q8 C22⋊Q8 C4⋊Q8 C22×Q8 C5×D4 C5×Q8 D4 Q8 C10 C2 # reps 1 3 1 6 3 2 4 12 4 24 12 8 4 4 16 16 1 4

Matrix representation of C5×D4×Q8 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 37 0 0 0 0 37
,
 40 2 0 0 40 1 0 0 0 0 1 0 0 0 0 1
,
 40 2 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 9 2 0 0 0 32
,
 40 0 0 0 0 40 0 0 0 0 7 6 0 0 19 34
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[40,40,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,2,32],[40,0,0,0,0,40,0,0,0,0,7,19,0,0,6,34] >;

C5×D4×Q8 in GAP, Magma, Sage, TeX

C_5\times D_4\times Q_8
% in TeX

G:=Group("C5xD4xQ8");
// GroupNames label

G:=SmallGroup(320,1551);
// by ID

G=gap.SmallGroup(320,1551);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,1242,304]);
// Polycyclic

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