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## G = (C2×Dic5)⋊3D4order 320 = 26·5

### 1st semidirect product of C2×Dic5 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×Dic5)⋊3D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — (C2×Dic5)⋊3D4
 Lower central C5 — C22×C10 — (C2×Dic5)⋊3D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×Dic5)⋊3D4
G = < a,b,c,d,e | a2=b10=d4=e2=1, c2=b5, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=b5c, ede=d-1 >

Subgroups: 1174 in 238 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.10D4, C10.D4, D10⋊C4, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C5×C2.C42, C2×C10.D4, C2×D10⋊C4, C22×D20, (C2×Dic5)⋊3D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C22.D4, C4.4D4, D20, C22×D5, C23.10D4, C2×D20, C4○D20, D4×D5, Q82D5, C4.D20, C22⋊D20, D10⋊D4, D10.13D4, C4⋊D20, (C2×Dic5)⋊3D4

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

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4F 4G 4H 4I 4J 5A 5B 10A ··· 10N 20A ··· 20X 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 4 ··· 4 20 20 20 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D20 C4○D20 D4×D5 Q8⋊2D5 kernel (C2×Dic5)⋊3D4 C5×C2.C42 C2×C10.D4 C2×D10⋊C4 C22×D20 C2×Dic5 C2×C20 C22×D5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 1 4 1 2 2 4 2 6 6 8 16 6 2

Matrix representation of (C2×Dic5)⋊3D4 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
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 6 1 0 0 0 0 5 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 2 0 0 0 0 40 1 0 0 0 0 0 0 28 32 0 0 0 0 28 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 23 0 0 0 0 0 32 0 0 0 0 0 0 2 9 0 0 0 0 4 39 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 6 1 0 0 0 0 6 35 0 0 0 0 0 0 1 0 0 0 0 0 0 40

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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,5,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,28,28,0,0,0,0,32,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,23,32,0,0,0,0,0,0,2,4,0,0,0,0,9,39,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×Dic5)⋊3D4 in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_5)\rtimes_3D_4
% in TeX

G:=Group("(C2xDic5):3D4");
// GroupNames label

G:=SmallGroup(320,299);
// by ID

G=gap.SmallGroup(320,299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,254,387,268,12550]);
// Polycyclic

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

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