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## G = C2×C4.D20order 320 = 26·5

### Direct product of C2 and C4.D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C4.D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D10⋊C4 — C2×C4.D20
 Lower central C5 — C2×C10 — C2×C4.D20
 Upper central C1 — C23 — C2×C42

Generators and relations for C2×C4.D20
G = < a,b,c,d | a2=b4=c20=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 1374 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×10], C2×C4 [×12], D4 [×8], Q8 [×8], C23, C23 [×16], D5 [×4], C10, C10 [×6], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C2×Q8 [×8], C24 [×2], Dic5 [×4], C20 [×4], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, Dic10 [×8], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×D5 [×4], C22×D5 [×12], C22×C10, C2×C4.4D4, D10⋊C4 [×16], C4×C20 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5 [×2], C4.D20 [×8], C2×D10⋊C4 [×4], C2×C4×C20, C22×Dic10, C22×D20, C2×C4.D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], D20 [×4], C22×D5 [×7], C2×C4.4D4, C2×D20 [×6], C4○D20 [×4], C23×D5, C4.D20 [×4], C22×D20, C2×C4○D20 [×2], C2×C4.D20

Smallest permutation representation of C2×C4.D20
On 160 points
Generators in S160
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(81 159)(82 160)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 121)(120 122)
(1 83 120 64)(2 84 101 65)(3 85 102 66)(4 86 103 67)(5 87 104 68)(6 88 105 69)(7 89 106 70)(8 90 107 71)(9 91 108 72)(10 92 109 73)(11 93 110 74)(12 94 111 75)(13 95 112 76)(14 96 113 77)(15 97 114 78)(16 98 115 79)(17 99 116 80)(18 100 117 61)(19 81 118 62)(20 82 119 63)(21 43 151 132)(22 44 152 133)(23 45 153 134)(24 46 154 135)(25 47 155 136)(26 48 156 137)(27 49 157 138)(28 50 158 139)(29 51 159 140)(30 52 160 121)(31 53 141 122)(32 54 142 123)(33 55 143 124)(34 56 144 125)(35 57 145 126)(36 58 146 127)(37 59 147 128)(38 60 148 129)(39 41 149 130)(40 42 150 131)
(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 30 120 160)(2 159 101 29)(3 28 102 158)(4 157 103 27)(5 26 104 156)(6 155 105 25)(7 24 106 154)(8 153 107 23)(9 22 108 152)(10 151 109 21)(11 40 110 150)(12 149 111 39)(13 38 112 148)(14 147 113 37)(15 36 114 146)(16 145 115 35)(17 34 116 144)(18 143 117 33)(19 32 118 142)(20 141 119 31)(41 75 130 94)(42 93 131 74)(43 73 132 92)(44 91 133 72)(45 71 134 90)(46 89 135 70)(47 69 136 88)(48 87 137 68)(49 67 138 86)(50 85 139 66)(51 65 140 84)(52 83 121 64)(53 63 122 82)(54 81 123 62)(55 61 124 100)(56 99 125 80)(57 79 126 98)(58 97 127 78)(59 77 128 96)(60 95 129 76)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M 4N 4O 4P 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 20 20 2 ··· 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D20 C4○D20 kernel C2×C4.D20 C4.D20 C2×D10⋊C4 C2×C4×C20 C22×Dic10 C22×D20 C2×C20 C2×C42 C2×C10 C42 C22×C4 C2×C4 C22 # reps 1 8 4 1 1 1 4 2 8 8 6 16 32

Matrix representation of C2×C4.D20 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 30 32 0 0 0 9 11 0 0 0 0 0 11 19 0 0 0 13 30
,
 1 0 0 0 0 0 22 9 0 0 0 32 0 0 0 0 0 0 3 21 0 0 0 23 24
,
 40 0 0 0 0 0 38 17 0 0 0 38 3 0 0 0 0 0 0 22 0 0 0 13 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,30,9,0,0,0,32,11,0,0,0,0,0,11,13,0,0,0,19,30],[1,0,0,0,0,0,22,32,0,0,0,9,0,0,0,0,0,0,3,23,0,0,0,21,24],[40,0,0,0,0,0,38,38,0,0,0,17,3,0,0,0,0,0,0,13,0,0,0,22,0] >;

C2×C4.D20 in GAP, Magma, Sage, TeX

C_2\times C_4.D_{20}
% in TeX

G:=Group("C2xC4.D20");
// GroupNames label

G:=SmallGroup(320,1148);
// by ID

G=gap.SmallGroup(320,1148);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,136,12550]);
// Polycyclic

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

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