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## G = C5×Q8.D4order 320 = 26·5

### Direct product of C5 and Q8.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×Q8.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — D4×C10 — C5×C4.4D4 — C5×Q8.D4
 Lower central C1 — C2 — C2×C4 — C5×Q8.D4
 Upper central C1 — C2×C10 — C4×C20 — C5×Q8.D4

Generators and relations for C5×Q8.D4
G = < a,b,c,d,e | a5=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 202 in 112 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, Q8.D4, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C5×Q16, D4×C10, Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C4⋊C8, Q8×C20, C5×C4.4D4, C10×SD16, C10×Q16, C5×Q8.D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C4○D8, C8.C22, C5×D4, C22×C10, Q8.D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C4○D8, C5×C8.C22, C5×Q8.D4

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

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E ··· 4I 4J 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M 10N 10O 10P 20A ··· 20P 20Q ··· 20AJ 20AK 20AL 20AM 20AN 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 ··· 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 8 2 2 2 2 4 ··· 4 8 1 1 1 1 4 4 4 4 1 ··· 1 8 8 8 8 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 D4 D4 C4○D4 C4○D8 C5×D4 C5×D4 C5×C4○D4 C5×C4○D8 C8.C22 C5×C8.C22 kernel C5×Q8.D4 C5×D4⋊C4 C5×Q8⋊C4 C5×C4⋊C8 Q8×C20 C5×C4.4D4 C10×SD16 C10×Q16 Q8.D4 D4⋊C4 Q8⋊C4 C4⋊C8 C4×Q8 C4.4D4 C2×SD16 C2×Q16 C2×C20 C5×Q8 C20 C10 C2×C4 Q8 C4 C2 C10 C2 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 2 2 2 4 8 8 8 16 1 4

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

 1 0 0 0 0 1 0 0 0 0 18 0 0 0 0 18
,
 1 39 0 0 1 40 0 0 0 0 40 0 0 0 0 40
,
 30 11 0 0 15 11 0 0 0 0 7 37 0 0 12 34
,
 9 0 0 0 0 9 0 0 0 0 19 36 0 0 15 22
,
 9 0 0 0 9 32 0 0 0 0 19 36 0 0 31 22
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[1,1,0,0,39,40,0,0,0,0,40,0,0,0,0,40],[30,15,0,0,11,11,0,0,0,0,7,12,0,0,37,34],[9,0,0,0,0,9,0,0,0,0,19,15,0,0,36,22],[9,9,0,0,0,32,0,0,0,0,19,31,0,0,36,22] >;

C5×Q8.D4 in GAP, Magma, Sage, TeX

C_5\times Q_8.D_4
% in TeX

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

G:=SmallGroup(320,965);
// by ID

G=gap.SmallGroup(320,965);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,856,10085,2539,124]);
// Polycyclic

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

׿
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