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## G = C2×Dic5.D4order 320 = 26·5

### Direct product of C2 and Dic5.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×Dic5.D4
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — C2×C22.F5 — C2×Dic5.D4
 Lower central C5 — C10 — C2×C10 — C2×Dic5.D4
 Upper central C1 — C22 — C23 — C22×C4

Generators and relations for C2×Dic5.D4
G = < a,b,c,d,e | a2=b10=1, c2=d4=b5, e2=c, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe-1=b3, dcd-1=b5c, ce=ec, ede-1=cd3 >

Subgroups: 522 in 146 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×12], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×Q8 [×8], Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C5⋊C8 [×4], Dic10 [×8], C2×Dic5 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C4.10D4, C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, Dic5.D4 [×4], C2×C22.F5 [×2], C22×Dic10, C2×Dic5.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4.10D4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4.10D4, C22⋊F5 [×2], C22×F5, Dic5.D4 [×2], C2×C22⋊F5, C2×Dic5.D4

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + - + + + - image C1 C2 C2 C2 C4 C4 C4 D4 F5 C4.10D4 C2×F5 C2×F5 C22⋊F5 Dic5.D4 kernel C2×Dic5.D4 Dic5.D4 C2×C22.F5 C22×Dic10 C2×Dic10 C22×Dic5 C22×C20 C2×Dic5 C22×C4 C10 C2×C4 C23 C22 C2 # reps 1 4 2 1 4 2 2 4 1 2 2 1 4 8

Matrix representation of C2×Dic5.D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 1 0 0 0 0 5 1 0 0 0 0 22 22 34 34 0 0 32 0 7 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 15 12 0 0 0 0 29 26 0 0 0 0 20 0 12 16 0 0 38 26 14 29
,
 23 15 0 0 0 0 14 18 0 0 0 0 0 0 7 1 20 20 0 0 3 18 0 23 0 0 28 2 17 40 0 0 24 31 17 40
,
 20 37 0 0 0 0 8 21 0 0 0 0 0 0 0 0 40 1 0 0 22 9 39 34 0 0 39 15 32 0 0 0 13 27 32 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,5,22,32,0,0,1,1,22,0,0,0,0,0,34,7,0,0,0,0,34,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,15,29,20,38,0,0,12,26,0,26,0,0,0,0,12,14,0,0,0,0,16,29],[23,14,0,0,0,0,15,18,0,0,0,0,0,0,7,3,28,24,0,0,1,18,2,31,0,0,20,0,17,17,0,0,20,23,40,40],[20,8,0,0,0,0,37,21,0,0,0,0,0,0,0,22,39,13,0,0,0,9,15,27,0,0,40,39,32,32,0,0,1,34,0,0] >;

C2×Dic5.D4 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic5.D4");
// GroupNames label

G:=SmallGroup(320,1098);
// by ID

G=gap.SmallGroup(320,1098);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,297,136,1684,6278,1595]);
// Polycyclic

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

׿
×
𝔽