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## G = C2×C20.53D4order 320 = 26·5

### Direct product of C2 and C20.53D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C20.53D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×C20.53D4
 Lower central C5 — C10 — C20 — C2×C20.53D4
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C2×C20.53D4
G = < a,b,c,d | a2=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b10c3 >

Subgroups: 238 in 106 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C20, C2×C10, C2×C10, C8.C4, C22×C8, C2×M4(2), C2×M4(2), C52C8, C52C8, C40, C2×C20, C22×C10, C2×C8.C4, C2×C52C8, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C20.53D4, C22×C52C8, C2×C4.Dic5, C10×M4(2), C2×C20.53D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C8.C4, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, C2×C8.C4, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C20.53D4, C2×C10.D4, C2×C20.53D4

Smallest permutation representation of C2×C20.53D4
On 160 points
Generators in S160
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 118)(22 119)(23 120)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 61)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
(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)
(1 127 89 157 11 137 99 147)(2 136 90 146 12 126 100 156)(3 125 91 155 13 135 81 145)(4 134 92 144 14 124 82 154)(5 123 93 153 15 133 83 143)(6 132 94 142 16 122 84 152)(7 121 95 151 17 131 85 141)(8 130 96 160 18 140 86 150)(9 139 97 149 19 129 87 159)(10 128 98 158 20 138 88 148)(21 43 103 69 31 53 113 79)(22 52 104 78 32 42 114 68)(23 41 105 67 33 51 115 77)(24 50 106 76 34 60 116 66)(25 59 107 65 35 49 117 75)(26 48 108 74 36 58 118 64)(27 57 109 63 37 47 119 73)(28 46 110 72 38 56 120 62)(29 55 111 61 39 45 101 71)(30 44 112 70 40 54 102 80)
(1 32 6 37 11 22 16 27)(2 21 7 26 12 31 17 36)(3 30 8 35 13 40 18 25)(4 39 9 24 14 29 19 34)(5 28 10 33 15 38 20 23)(41 143 46 148 51 153 56 158)(42 152 47 157 52 142 57 147)(43 141 48 146 53 151 58 156)(44 150 49 155 54 160 59 145)(45 159 50 144 55 149 60 154)(61 139 66 124 71 129 76 134)(62 128 67 133 72 138 77 123)(63 137 68 122 73 127 78 132)(64 126 69 131 74 136 79 121)(65 135 70 140 75 125 80 130)(81 112 86 117 91 102 96 107)(82 101 87 106 92 111 97 116)(83 110 88 115 93 120 98 105)(84 119 89 104 94 109 99 114)(85 108 90 113 95 118 100 103)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 8 ··· 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + - - + + + - - image C1 C2 C2 C2 C2 C4 D4 Q8 Q8 D5 D10 D10 C8.C4 Dic10 C4×D5 C5⋊D4 Dic10 C20.53D4 kernel C2×C20.53D4 C20.53D4 C22×C5⋊2C8 C2×C4.Dic5 C10×M4(2) C2×C5⋊2C8 C2×C20 C2×C20 C22×C10 C2×M4(2) M4(2) C22×C4 C10 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 8 2 1 1 2 4 2 8 4 8 8 4 8

Matrix representation of C2×C20.53D4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 35 36 0 0 40 40 0 0 0 0 9 0 0 0 0 9
,
 27 17 0 0 27 14 0 0 0 0 3 0 0 0 0 14
,
 18 10 0 0 21 23 0 0 0 0 0 27 0 0 14 0
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[35,40,0,0,36,40,0,0,0,0,9,0,0,0,0,9],[27,27,0,0,17,14,0,0,0,0,3,0,0,0,0,14],[18,21,0,0,10,23,0,0,0,0,0,14,0,0,27,0] >;

C2×C20.53D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{53}D_4
% in TeX

G:=Group("C2xC20.53D4");
// GroupNames label

G:=SmallGroup(320,750);
// by ID

G=gap.SmallGroup(320,750);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,58,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^10*c^3>;
// generators/relations

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