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## G = C5×D4.7D4order 320 = 26·5

### Direct product of C5 and D4.7D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.7D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — Q8×C10 — C10×SD16 — C5×D4.7D4
 Lower central C1 — C2 — C2×C4 — C5×D4.7D4
 Upper central C1 — C2×C10 — C22×C20 — C5×D4.7D4

Generators and relations for C5×D4.7D4
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >

Subgroups: 274 in 152 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, D4.7D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C5×Q16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊C8, C5×D4⋊C4, C5×Q8⋊C4, C5×C22⋊Q8, C10×SD16, C10×Q16, C10×C4○D4, C5×D4.7D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C4○D8, C8.C22, C5×D4, C22×C10, D4.7D4, D4×C10, C5×C22≀C2, C5×C4○D8, C5×C8.C22, C5×D4.7D4

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

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10X 20A ··· 20P 20Q ··· 20X 20Y ··· 20AF 40A ··· 40P order 1 2 2 2 2 2 2 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 4 2 2 2 2 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 1 1 1 1 2 2 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 D4 D4 C4○D8 C5×D4 C5×D4 C5×D4 C5×D4 C5×C4○D8 C8.C22 C5×C8.C22 kernel C5×D4.7D4 C5×C22⋊C8 C5×D4⋊C4 C5×Q8⋊C4 C5×C22⋊Q8 C10×SD16 C10×Q16 C10×C4○D4 D4.7D4 C22⋊C8 D4⋊C4 Q8⋊C4 C22⋊Q8 C2×SD16 C2×Q16 C2×C4○D4 C2×C20 C5×D4 C5×Q8 C22×C10 C10 C2×C4 D4 Q8 C23 C2 C10 C2 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 2 2 1 4 4 8 8 4 16 1 4

Matrix representation of C5×D4.7D4 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 18 0 0 0 0 18
,
 9 0 0 0 0 32 0 0 0 0 1 0 0 0 0 1
,
 0 32 0 0 9 0 0 0 0 0 40 0 0 0 0 40
,
 0 38 0 0 27 0 0 0 0 0 1 39 0 0 1 40
,
 0 14 0 0 38 0 0 0 0 0 1 39 0 0 0 40
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[9,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,32,0,0,0,0,0,40,0,0,0,0,40],[0,27,0,0,38,0,0,0,0,0,1,1,0,0,39,40],[0,38,0,0,14,0,0,0,0,0,1,0,0,0,39,40] >;

C5×D4.7D4 in GAP, Magma, Sage, TeX

C_5\times D_4._7D_4
% in TeX

G:=Group("C5xD4.7D4");
// GroupNames label

G:=SmallGroup(320,953);
// by ID

G=gap.SmallGroup(320,953);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,856,7004,3511,172]);
// Polycyclic

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

׿
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𝔽