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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 — C22 — C2×C4 — C2×C20 — D4×C10 — C10×SD16 — C5×Q8⋊D4
 Lower central C1 — C2 — C2×C4 — C5×Q8⋊D4
 Upper central C1 — C2×C10 — C22×C20 — C5×Q8⋊D4

Generators and relations for C5×Q8⋊D4
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=ebe=b-1, dcd-1=ece=b-1c, ede=d-1 >

Subgroups: 290 in 158 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, Q8⋊D4, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, Q8×C10, C5×C22⋊C8, C5×Q8⋊C4, C5×C4⋊D4, C10×SD16, Q8×C2×C10, C5×Q8⋊D4
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C2×C10, C22≀C2, C2×SD16, C8.C22, C5×D4, C22×C10, Q8⋊D4, C5×SD16, D4×C10, C5×C22≀C2, C10×SD16, 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 47 12 28)(2 48 13 29)(3 49 14 30)(4 50 15 26)(5 46 11 27)(6 152 144 31)(7 153 145 32)(8 154 141 33)(9 155 142 34)(10 151 143 35)(16 25 137 158)(17 21 138 159)(18 22 139 160)(19 23 140 156)(20 24 136 157)(36 44 70 53)(37 45 66 54)(38 41 67 55)(39 42 68 51)(40 43 69 52)(56 106 94 77)(57 107 95 78)(58 108 91 79)(59 109 92 80)(60 110 93 76)(61 74 82 88)(62 75 83 89)(63 71 84 90)(64 72 85 86)(65 73 81 87)(96 117 134 146)(97 118 135 147)(98 119 131 148)(99 120 132 149)(100 116 133 150)(101 128 122 114)(102 129 123 115)(103 130 124 111)(104 126 125 112)(105 127 121 113)
(1 157 12 24)(2 158 13 25)(3 159 14 21)(4 160 15 22)(5 156 11 23)(6 70 144 36)(7 66 145 37)(8 67 141 38)(9 68 142 39)(10 69 143 40)(16 48 137 29)(17 49 138 30)(18 50 139 26)(19 46 140 27)(20 47 136 28)(31 53 152 44)(32 54 153 45)(33 55 154 41)(34 51 155 42)(35 52 151 43)(56 127 94 113)(57 128 95 114)(58 129 91 115)(59 130 92 111)(60 126 93 112)(61 147 82 118)(62 148 83 119)(63 149 84 120)(64 150 85 116)(65 146 81 117)(71 132 90 99)(72 133 86 100)(73 134 87 96)(74 135 88 97)(75 131 89 98)(76 125 110 104)(77 121 106 105)(78 122 107 101)(79 123 108 102)(80 124 109 103)
(1 81 66 106)(2 82 67 107)(3 83 68 108)(4 84 69 109)(5 85 70 110)(6 112 23 100)(7 113 24 96)(8 114 25 97)(9 115 21 98)(10 111 22 99)(11 64 36 76)(12 65 37 77)(13 61 38 78)(14 62 39 79)(15 63 40 80)(16 118 33 101)(17 119 34 102)(18 120 35 103)(19 116 31 104)(20 117 32 105)(26 90 43 92)(27 86 44 93)(28 87 45 94)(29 88 41 95)(30 89 42 91)(46 72 53 60)(47 73 54 56)(48 74 55 57)(49 75 51 58)(50 71 52 59)(121 136 146 153)(122 137 147 154)(123 138 148 155)(124 139 149 151)(125 140 150 152)(126 156 133 144)(127 157 134 145)(128 158 135 141)(129 159 131 142)(130 160 132 143)
(6 152)(7 153)(8 154)(9 155)(10 151)(16 158)(17 159)(18 160)(19 156)(20 157)(21 138)(22 139)(23 140)(24 136)(25 137)(26 50)(27 46)(28 47)(29 48)(30 49)(31 144)(32 145)(33 141)(34 142)(35 143)(41 55)(42 51)(43 52)(44 53)(45 54)(56 87)(57 88)(58 89)(59 90)(60 86)(61 78)(62 79)(63 80)(64 76)(65 77)(71 92)(72 93)(73 94)(74 95)(75 91)(81 106)(82 107)(83 108)(84 109)(85 110)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 10U 10V 10W 10X 20A ··· 20H 20I ··· 20AB 20AC 20AD 20AE 20AF 40A ··· 40P order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 8 2 2 4 ··· 4 8 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 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 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D4 SD16 C5×D4 C5×D4 C5×D4 C5×SD16 C8.C22 C5×C8.C22 kernel C5×Q8⋊D4 C5×C22⋊C8 C5×Q8⋊C4 C5×C4⋊D4 C10×SD16 Q8×C2×C10 Q8⋊D4 C22⋊C8 Q8⋊C4 C4⋊D4 C2×SD16 C22×Q8 C2×C20 C5×Q8 C22×C10 C2×C10 C2×C4 Q8 C23 C22 C10 C2 # reps 1 1 2 1 2 1 4 4 8 4 8 4 1 4 1 4 4 16 4 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
,
 0 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 26 26 0 0 26 15 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 1 0 0 0 0 0 1 0 0 40 0
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[26,26,0,0,26,15,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1] >;

C5×Q8⋊D4 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes D_4
% in TeX

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

G:=SmallGroup(320,949);
// by ID

G=gap.SmallGroup(320,949);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1128,1766,7004,3511,172]);
// 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=e*b*e=b^-1,d*c*d^-1=e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
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𝔽