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## G = C10×D4⋊C4order 320 = 26·5

### Direct product of C10 and D4⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C10×D4⋊C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 402 in 202 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C20, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, C2×D4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C22×C20, C22×C20, D4×C10, D4×C10, C23×C10, C5×D4⋊C4, C10×C4⋊C4, C22×C40, D4×C2×C10, C10×D4⋊C4
Quotients:

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

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

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

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

140 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A ··· 8H 10A ··· 10AB 10AC ··· 10AR 20A ··· 20P 20Q ··· 20AF 40A ··· 40AF order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 ··· 1 4 4 4 4 2 2 2 2 4 4 4 4 1 1 1 1 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 D4 D4 D8 SD16 C5×D4 C5×D4 C5×D8 C5×SD16 kernel C10×D4⋊C4 C5×D4⋊C4 C10×C4⋊C4 C22×C40 D4×C2×C10 D4×C10 C2×D4⋊C4 D4⋊C4 C2×C4⋊C4 C22×C8 C22×D4 C2×D4 C2×C20 C22×C10 C2×C10 C2×C10 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 4 16 4 4 4 32 3 1 4 4 12 4 16 16

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

 40 0 0 0 0 25 0 0 0 0 25 0 0 0 0 25
,
 1 0 0 0 0 1 0 0 0 0 1 39 0 0 1 40
,
 40 0 0 0 0 1 0 0 0 0 1 39 0 0 0 40
,
 1 0 0 0 0 32 0 0 0 0 0 17 0 0 29 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,39,40],[1,0,0,0,0,32,0,0,0,0,0,29,0,0,17,0] >;

C10×D4⋊C4 in GAP, Magma, Sage, TeX

C_{10}\times D_4\rtimes C_4
% in TeX

G:=Group("C10xD4:C4");
// GroupNames label

G:=SmallGroup(320,915);
// by ID

G=gap.SmallGroup(320,915);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172]);
// Polycyclic

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

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