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## G = C4○D4×C2×C10order 320 = 26·5

### Direct product of C2×C10 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C2×C10
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — C5×C4○D4 — C10×C4○D4 — C4○D4×C2×C10
 Lower central C1 — C2 — C4○D4×C2×C10
 Upper central C1 — C22×C20 — C4○D4×C2×C10

Generators and relations for C4○D4×C2×C10
G = < a,b,c,d,e | a2=b10=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1010 in 890 conjugacy classes, 770 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C20, C2×C10, C2×C10, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C22×C4○D4, C22×C20, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C23×C10, C23×C20, D4×C2×C10, Q8×C2×C10, C10×C4○D4, C4○D4×C2×C10
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, C25, C22×C10, C22×C4○D4, C5×C4○D4, C23×C10, C10×C4○D4, C24×C10, C4○D4×C2×C10

Smallest permutation representation of C4○D4×C2×C10
On 160 points
Generators in S160
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 41)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 131)(19 132)(20 133)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 31)(28 32)(29 33)(30 34)(51 87)(52 88)(53 89)(54 90)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 74)(91 127)(92 128)(93 129)(94 130)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 111)(108 112)(109 113)(110 114)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 151)(148 152)(149 153)(150 154)
(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 87 27 76)(2 88 28 77)(3 89 29 78)(4 90 30 79)(5 81 21 80)(6 82 22 71)(7 83 23 72)(8 84 24 73)(9 85 25 74)(10 86 26 75)(11 110 159 99)(12 101 160 100)(13 102 151 91)(14 103 152 92)(15 104 153 93)(16 105 154 94)(17 106 155 95)(18 107 156 96)(19 108 157 97)(20 109 158 98)(31 62 42 51)(32 63 43 52)(33 64 44 53)(34 65 45 54)(35 66 46 55)(36 67 47 56)(37 68 48 57)(38 69 49 58)(39 70 50 59)(40 61 41 60)(111 142 122 131)(112 143 123 132)(113 144 124 133)(114 145 125 134)(115 146 126 135)(116 147 127 136)(117 148 128 137)(118 149 129 138)(119 150 130 139)(120 141 121 140)
(1 127 27 116)(2 128 28 117)(3 129 29 118)(4 130 30 119)(5 121 21 120)(6 122 22 111)(7 123 23 112)(8 124 24 113)(9 125 25 114)(10 126 26 115)(11 70 159 59)(12 61 160 60)(13 62 151 51)(14 63 152 52)(15 64 153 53)(16 65 154 54)(17 66 155 55)(18 67 156 56)(19 68 157 57)(20 69 158 58)(31 102 42 91)(32 103 43 92)(33 104 44 93)(34 105 45 94)(35 106 46 95)(36 107 47 96)(37 108 48 97)(38 109 49 98)(39 110 50 99)(40 101 41 100)(71 142 82 131)(72 143 83 132)(73 144 84 133)(74 145 85 134)(75 146 86 135)(76 147 87 136)(77 148 88 137)(78 149 89 138)(79 150 90 139)(80 141 81 140)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 129)(30 130)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)

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

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

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

200 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4H 4I ··· 4T 5A 5B 5C 5D 10A ··· 10AB 10AC ··· 10BX 20A ··· 20AF 20AG ··· 20CB order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 C4○D4 C5×C4○D4 kernel C4○D4×C2×C10 C23×C20 D4×C2×C10 Q8×C2×C10 C10×C4○D4 C22×C4○D4 C23×C4 C22×D4 C22×Q8 C2×C4○D4 C2×C10 C22 # reps 1 3 3 1 24 4 12 12 4 96 8 32

Matrix representation of C4○D4×C2×C10 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 37 0 0 0 0 37
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
 40 0 0 0 0 1 0 0 0 0 1 39 0 0 1 40
,
 1 0 0 0 0 40 0 0 0 0 1 39 0 0 0 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,39,40] >;

C4○D4×C2×C10 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_2\times C_{10}
% in TeX

G:=Group("C4oD4xC2xC10");
// GroupNames label

G:=SmallGroup(320,1631);
// by ID

G=gap.SmallGroup(320,1631);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269,856]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^4=e^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations

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𝔽