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## G = C5×D4⋊2Q8order 320 = 26·5

### Direct product of C5 and D4⋊2Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4⋊2Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×C4⋊Q8 — C5×D4⋊2Q8
 Lower central C1 — C2 — C2×C4 — C5×D4⋊2Q8
 Upper central C1 — C2×C10 — C4×C20 — C5×D4⋊2Q8

Generators and relations for C5×D42Q8
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >

Subgroups: 202 in 108 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D42Q8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, D4×C10, Q8×C10, C5×D4⋊C4, C5×C4⋊C8, C5×C4.Q8, D4×C20, C5×C4⋊Q8, C5×D42Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, SD16, C2×D4, C2×Q8, C4○D4, C2×C10, C22⋊Q8, C2×SD16, C8⋊C22, C5×D4, C5×Q8, C22×C10, D42Q8, C5×SD16, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C10×SD16, C5×C8⋊C22, C5×D42Q8

Smallest permutation representation of C5×D42Q8
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 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 23 144 156)(7 24 145 157)(8 25 141 158)(9 21 142 159)(10 22 143 160)(16 33 137 154)(17 34 138 155)(18 35 139 151)(19 31 140 152)(20 32 136 153)(26 43 50 52)(27 44 46 53)(28 45 47 54)(29 41 48 55)(30 42 49 51)(56 87 94 73)(57 88 95 74)(58 89 91 75)(59 90 92 71)(60 86 93 72)(61 107 82 78)(62 108 83 79)(63 109 84 80)(64 110 85 76)(65 106 81 77)(96 113 134 127)(97 114 135 128)(98 115 131 129)(99 111 132 130)(100 112 133 126)(101 118 122 147)(102 119 123 148)(103 120 124 149)(104 116 125 150)(105 117 121 146)
(1 47)(2 48)(3 49)(4 50)(5 46)(6 140)(7 136)(8 137)(9 138)(10 139)(11 27)(12 28)(13 29)(14 30)(15 26)(16 141)(17 142)(18 143)(19 144)(20 145)(21 34)(22 35)(23 31)(24 32)(25 33)(36 44)(37 45)(38 41)(39 42)(40 43)(51 68)(52 69)(53 70)(54 66)(55 67)(56 77)(57 78)(58 79)(59 80)(60 76)(61 74)(62 75)(63 71)(64 72)(65 73)(81 87)(82 88)(83 89)(84 90)(85 86)(91 108)(92 109)(93 110)(94 106)(95 107)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 120)(112 116)(113 117)(114 118)(115 119)(126 150)(127 146)(128 147)(129 148)(130 149)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 87 54 106)(2 88 55 107)(3 89 51 108)(4 90 52 109)(5 86 53 110)(6 112 19 116)(7 113 20 117)(8 114 16 118)(9 115 17 119)(10 111 18 120)(11 72 44 76)(12 73 45 77)(13 74 41 78)(14 75 42 79)(15 71 43 80)(21 131 34 123)(22 132 35 124)(23 133 31 125)(24 134 32 121)(25 135 33 122)(26 84 40 92)(27 85 36 93)(28 81 37 94)(29 82 38 95)(30 83 39 91)(46 64 70 60)(47 65 66 56)(48 61 67 57)(49 62 68 58)(50 63 69 59)(96 153 105 157)(97 154 101 158)(98 155 102 159)(99 151 103 160)(100 152 104 156)(126 140 150 144)(127 136 146 145)(128 137 147 141)(129 138 148 142)(130 139 149 143)
(1 134 54 121)(2 135 55 122)(3 131 51 123)(4 132 52 124)(5 133 53 125)(6 85 19 93)(7 81 20 94)(8 82 16 95)(9 83 17 91)(10 84 18 92)(11 100 44 104)(12 96 45 105)(13 97 41 101)(14 98 42 102)(15 99 43 103)(21 108 34 89)(22 109 35 90)(23 110 31 86)(24 106 32 87)(25 107 33 88)(26 120 40 111)(27 116 36 112)(28 117 37 113)(29 118 38 114)(30 119 39 115)(46 150 70 126)(47 146 66 127)(48 147 67 128)(49 148 68 129)(50 149 69 130)(56 145 65 136)(57 141 61 137)(58 142 62 138)(59 143 63 139)(60 144 64 140)(71 160 80 151)(72 156 76 152)(73 157 77 153)(74 158 78 154)(75 159 79 155)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AB 20AC ··· 20AJ 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 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 Q8 SD16 C4○D4 C5×D4 C5×Q8 C5×SD16 C5×C4○D4 C8⋊C22 C5×C8⋊C22 kernel C5×D4⋊2Q8 C5×D4⋊C4 C5×C4⋊C8 C5×C4.Q8 D4×C20 C5×C4⋊Q8 D4⋊2Q8 D4⋊C4 C4⋊C8 C4.Q8 C4×D4 C4⋊Q8 C2×C20 C5×D4 C20 C20 C2×C4 D4 C4 C4 C10 C2 # reps 1 2 1 2 1 1 4 8 4 8 4 4 2 2 4 2 8 8 16 8 1 4

Matrix representation of C5×D42Q8 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 40 0
,
 1 0 0 0 21 40 0 0 0 0 0 40 0 0 40 0
,
 32 0 0 0 16 9 0 0 0 0 0 1 0 0 40 0
,
 21 39 0 0 16 20 0 0 0 0 26 15 0 0 15 15
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,21,0,0,0,40,0,0,0,0,0,40,0,0,40,0],[32,16,0,0,0,9,0,0,0,0,0,40,0,0,1,0],[21,16,0,0,39,20,0,0,0,0,26,15,0,0,15,15] >;

C5×D42Q8 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_2Q_8
% in TeX

G:=Group("C5xD4:2Q8");
// GroupNames label

G:=SmallGroup(320,977);
// by ID

G=gap.SmallGroup(320,977);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1400,589,288,1766,10085,2539,124]);
// 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=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽