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## G = C2×D4.Dic5order 320 = 26·5

### Direct product of C2 and D4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4.Dic5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×D4.Dic5
 Lower central C5 — C10 — C2×D4.Dic5
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for C2×D4.Dic5
G = < a,b,c,d,e | a2=b4=1, c2=d10=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

Subgroups: 494 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C52C8, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C2×C8○D4, C2×C52C8, C2×C52C8, C4.Dic5, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C22×C52C8, C2×C4.Dic5, D4.Dic5, C10×C4○D4, C2×D4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C8○D4, C23×C4, C2×Dic5, C22×D5, C2×C8○D4, C22×Dic5, C23×D5, D4.Dic5, C23×Dic5, C2×D4.Dic5

Smallest permutation representation of C2×D4.Dic5
On 160 points
Generators in S160
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 101)(99 102)(100 103)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 141)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 151)
(1 46 11 56)(2 47 12 57)(3 48 13 58)(4 49 14 59)(5 50 15 60)(6 51 16 41)(7 52 17 42)(8 53 18 43)(9 54 19 44)(10 55 20 45)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 71 36 61)(27 72 37 62)(28 73 38 63)(29 74 39 64)(30 75 40 65)(81 132 91 122)(82 133 92 123)(83 134 93 124)(84 135 94 125)(85 136 95 126)(86 137 96 127)(87 138 97 128)(88 139 98 129)(89 140 99 130)(90 121 100 131)(101 160 111 150)(102 141 112 151)(103 142 113 152)(104 143 114 153)(105 144 115 154)(106 145 116 155)(107 146 117 156)(108 147 118 157)(109 148 119 158)(110 149 120 159)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)(121 136 131 126)(122 137 132 127)(123 138 133 128)(124 139 134 129)(125 140 135 130)(141 156 151 146)(142 157 152 147)(143 158 153 148)(144 159 154 149)(145 160 155 150)
(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)
(1 118 6 103 11 108 16 113)(2 107 7 112 12 117 17 102)(3 116 8 101 13 106 18 111)(4 105 9 110 14 115 19 120)(5 114 10 119 15 104 20 109)(21 95 26 100 31 85 36 90)(22 84 27 89 32 94 37 99)(23 93 28 98 33 83 38 88)(24 82 29 87 34 92 39 97)(25 91 30 96 35 81 40 86)(41 152 46 157 51 142 56 147)(42 141 47 146 52 151 57 156)(43 150 48 155 53 160 58 145)(44 159 49 144 54 149 59 154)(45 148 50 153 55 158 60 143)(61 121 66 126 71 131 76 136)(62 130 67 135 72 140 77 125)(63 139 68 124 73 129 78 134)(64 128 69 133 74 138 79 123)(65 137 70 122 75 127 80 132)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 5A 5B 8A ··· 8H 8I ··· 8T 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 2 5 ··· 5 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + - - - + image C1 C2 C2 C2 C2 C4 C4 C4 D5 D10 Dic5 Dic5 Dic5 D10 C8○D4 D4.Dic5 kernel C2×D4.Dic5 C22×C5⋊2C8 C2×C4.Dic5 D4.Dic5 C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C4○D4 C10 C2 # reps 1 3 3 8 1 6 2 8 2 6 6 2 8 8 8 8

Matrix representation of C2×D4.Dic5 in GL4(𝔽41) generated by

 40 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 0 9 0 0 9 0
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 32
,
 40 40 0 0 8 7 0 0 0 0 9 0 0 0 0 9
,
 9 25 0 0 5 32 0 0 0 0 3 0 0 0 0 3
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[40,8,0,0,40,7,0,0,0,0,9,0,0,0,0,9],[9,5,0,0,25,32,0,0,0,0,3,0,0,0,0,3] >;

C2×D4.Dic5 in GAP, Magma, Sage, TeX

C_2\times D_4.{\rm Dic}_5
% in TeX

G:=Group("C2xD4.Dic5");
// GroupNames label

G:=SmallGroup(320,1490);
// by ID

G=gap.SmallGroup(320,1490);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,102,12550]);
// Polycyclic

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

׿
×
𝔽