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## G = C5×Q8⋊6D4order 320 = 26·5

### Direct product of C5 and Q8⋊6D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×Q8⋊6D4
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C4⋊D4 — C5×Q8⋊6D4
 Lower central C1 — C22 — C5×Q8⋊6D4
 Upper central C1 — C2×C10 — C5×Q8⋊6D4

Generators and relations for C5×Q86D4
G = < a,b,c,d,e | a5=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 506 in 312 conjugacy classes, 166 normal (20 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×5], C22, C22 [×18], C5, C2×C4, C2×C4 [×8], C2×C4 [×12], D4 [×24], Q8 [×4], C23 [×6], C10 [×3], C10 [×6], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C20 [×8], C20 [×5], C2×C10, C2×C10 [×18], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C41D4 [×3], C2×C4○D4 [×2], C2×C20, C2×C20 [×8], C2×C20 [×12], C5×D4 [×24], C5×Q8 [×4], C22×C10 [×6], Q86D4, C4×C20 [×3], C5×C22⋊C4 [×6], C5×C4⋊C4, C5×C4⋊C4 [×3], C22×C20 [×6], D4×C10 [×15], Q8×C10, C5×C4○D4 [×8], D4×C20 [×3], Q8×C20, C5×C4⋊D4 [×6], C5×C41D4 [×3], C10×C4○D4 [×2], C5×Q86D4
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×D4, C2×C4○D4, 2+ 1+4, C5×D4 [×4], C22×C10 [×15], Q86D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2+ 1+4, C5×Q86D4

Smallest permutation representation of C5×Q86D4
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 47 35 41)(2 48 31 42)(3 49 32 43)(4 50 33 44)(5 46 34 45)(6 146 160 145)(7 147 156 141)(8 148 157 142)(9 149 158 143)(10 150 159 144)(11 137 17 151)(12 138 18 152)(13 139 19 153)(14 140 20 154)(15 136 16 155)(21 55 27 36)(22 51 28 37)(23 52 29 38)(24 53 30 39)(25 54 26 40)(56 87 75 81)(57 88 71 82)(58 89 72 83)(59 90 73 84)(60 86 74 85)(61 95 67 76)(62 91 68 77)(63 92 69 78)(64 93 70 79)(65 94 66 80)(96 127 115 121)(97 128 111 122)(98 129 112 123)(99 130 113 124)(100 126 114 125)(101 135 107 116)(102 131 108 117)(103 132 109 118)(104 133 110 119)(105 134 106 120)
(1 81 35 87)(2 82 31 88)(3 83 32 89)(4 84 33 90)(5 85 34 86)(6 106 160 105)(7 107 156 101)(8 108 157 102)(9 109 158 103)(10 110 159 104)(11 97 17 111)(12 98 18 112)(13 99 19 113)(14 100 20 114)(15 96 16 115)(21 76 27 95)(22 77 28 91)(23 78 29 92)(24 79 30 93)(25 80 26 94)(36 61 55 67)(37 62 51 68)(38 63 52 69)(39 64 53 70)(40 65 54 66)(41 56 47 75)(42 57 48 71)(43 58 49 72)(44 59 50 73)(45 60 46 74)(116 147 135 141)(117 148 131 142)(118 149 132 143)(119 150 133 144)(120 146 134 145)(121 155 127 136)(122 151 128 137)(123 152 129 138)(124 153 130 139)(125 154 126 140)
(1 135 27 121)(2 131 28 122)(3 132 29 123)(4 133 30 124)(5 134 26 125)(6 66 14 60)(7 67 15 56)(8 68 11 57)(9 69 12 58)(10 70 13 59)(16 75 156 61)(17 71 157 62)(18 72 158 63)(19 73 159 64)(20 74 160 65)(21 127 35 116)(22 128 31 117)(23 129 32 118)(24 130 33 119)(25 126 34 120)(36 115 47 101)(37 111 48 102)(38 112 49 103)(39 113 50 104)(40 114 46 105)(41 107 55 96)(42 108 51 97)(43 109 52 98)(44 110 53 99)(45 106 54 100)(76 155 87 141)(77 151 88 142)(78 152 89 143)(79 153 90 144)(80 154 86 145)(81 147 95 136)(82 148 91 137)(83 149 92 138)(84 150 93 139)(85 146 94 140)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 94)(7 95)(8 91)(9 92)(10 93)(11 82)(12 83)(13 84)(14 85)(15 81)(16 87)(17 88)(18 89)(19 90)(20 86)(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)

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

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

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

125 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 4M 4N 4O 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AJ 20A ··· 20AV 20AW ··· 20BH order 1 2 2 2 2 ··· 2 4 ··· 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 ··· 4 2 ··· 2 4 4 4 1 1 1 1 1 ··· 1 4 ··· 4 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 D4 C4○D4 C5×D4 C5×C4○D4 2+ 1+4 C5×2+ 1+4 kernel C5×Q8⋊6D4 D4×C20 Q8×C20 C5×C4⋊D4 C5×C4⋊1D4 C10×C4○D4 Q8⋊6D4 C4×D4 C4×Q8 C4⋊D4 C4⋊1D4 C2×C4○D4 C5×Q8 C20 Q8 C4 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×Q86D4 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 18 0 0 0 0 18
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 40 0 0 0 0 40 0 0 0 0 0 9 0 0 9 0
,
 5 39 0 0 13 36 0 0 0 0 9 0 0 0 0 32
,
 36 2 0 0 29 5 0 0 0 0 0 32 0 0 9 0
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[5,13,0,0,39,36,0,0,0,0,9,0,0,0,0,32],[36,29,0,0,2,5,0,0,0,0,0,9,0,0,32,0] >;

C5×Q86D4 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_6D_4
% in TeX

G:=Group("C5xQ8:6D4");
// GroupNames label

G:=SmallGroup(320,1552);
// by ID

G=gap.SmallGroup(320,1552);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,436,1242,304]);
// Polycyclic

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

׿
×
𝔽