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## G = C2×D20⋊8C4order 320 = 26·5

### Direct product of C2 and D20⋊8C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D20⋊8C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×D20⋊8C4
 Lower central C5 — C10 — C2×D20⋊8C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×D208C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b10c >

Subgroups: 1518 in 426 conjugacy classes, 175 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×32], C5, C2×C4 [×10], C2×C4 [×30], D4 [×16], C23, C23 [×20], D5 [×8], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×18], C2×D4 [×12], C24 [×2], Dic5 [×4], Dic5 [×2], C20 [×4], C20 [×4], D10 [×8], D10 [×24], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×D5 [×16], D20 [×16], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×10], C2×C20 [×4], C22×D5 [×12], C22×D5 [×8], C22×C10, C2×C4×D4, C4×Dic5 [×4], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×C4×D5 [×8], C2×C4×D5 [×8], C2×D20 [×12], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5 [×2], D208C4 [×8], C2×C4×Dic5, C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4 [×2], C22×D20, C2×D208C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], D4×D5 [×2], Q82D5 [×2], C23×D5, D208C4 [×4], D5×C22×C4, C2×D4×D5, C2×Q82D5, C2×D208C4

Smallest permutation representation of C2×D208C4
On 160 points
Generators in S160
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 140)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)(81 85)(82 84)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(116 120)(117 119)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 139 101 81)(2 130 102 92)(3 121 103 83)(4 132 104 94)(5 123 105 85)(6 134 106 96)(7 125 107 87)(8 136 108 98)(9 127 109 89)(10 138 110 100)(11 129 111 91)(12 140 112 82)(13 131 113 93)(14 122 114 84)(15 133 115 95)(16 124 116 86)(17 135 117 97)(18 126 118 88)(19 137 119 99)(20 128 120 90)(21 72 155 45)(22 63 156 56)(23 74 157 47)(24 65 158 58)(25 76 159 49)(26 67 160 60)(27 78 141 51)(28 69 142 42)(29 80 143 53)(30 71 144 44)(31 62 145 55)(32 73 146 46)(33 64 147 57)(34 75 148 48)(35 66 149 59)(36 77 150 50)(37 68 151 41)(38 79 152 52)(39 70 153 43)(40 61 154 54)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4L 4M ··· 4T 4U 4V 4W 4X 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 ··· 10 2 ··· 2 5 ··· 5 10 10 10 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C4×D5 D4×D5 Q8⋊2D5 kernel C2×D20⋊8C4 D20⋊8C4 C2×C4×Dic5 C2×D10⋊C4 C10×C4⋊C4 D5×C22×C4 C22×D20 C2×D20 C2×Dic5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 1 2 1 2 1 16 4 2 4 8 6 16 4 4

Matrix representation of C2×D208C4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 34 40 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 9 0
,
 40 0 0 0 0 0 34 40 0 0 0 7 7 0 0 0 0 0 40 0 0 0 0 0 1
,
 1 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 1 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,9,0],[40,0,0,0,0,0,34,7,0,0,0,40,7,0,0,0,0,0,40,0,0,0,0,0,1],[1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,40,0] >;

C2×D208C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes_8C_4
% in TeX

G:=Group("C2xD20:8C4");
// GroupNames label

G:=SmallGroup(320,1175);
// by ID

G=gap.SmallGroup(320,1175);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,297,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^10*c>;
// generators/relations

׿
×
𝔽