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G = C23.14D20order 320 = 26·5

7th non-split extension by C23 of D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — C23.14D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4⋊Dic5 — C23.14D20
 Lower central C5 — C22×C10 — C23.14D20
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C23.14D20
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 566 in 170 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.11D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C10.10C42, C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C23.14D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22.D4, C4.4D4, C422C2, D20, C5⋊D4, C22×D5, C23.11D4, C2×D20, C4○D20, D42D5, C2×C5⋊D4, C23.D10, C22.D20, C207D4, C23.18D10, C20.17D4, C23.14D20

Smallest permutation representation of C23.14D20
On 160 points
Generators in S160
```(1 11)(2 140)(3 13)(4 122)(5 15)(6 124)(7 17)(8 126)(9 19)(10 128)(12 130)(14 132)(16 134)(18 136)(20 138)(21 72)(22 118)(23 74)(24 120)(25 76)(26 102)(27 78)(28 104)(29 80)(30 106)(31 62)(32 108)(33 64)(34 110)(35 66)(36 112)(37 68)(38 114)(39 70)(40 116)(41 107)(42 63)(43 109)(44 65)(45 111)(46 67)(47 113)(48 69)(49 115)(50 71)(51 117)(52 73)(53 119)(54 75)(55 101)(56 77)(57 103)(58 79)(59 105)(60 61)(81 91)(82 151)(83 93)(84 153)(85 95)(86 155)(87 97)(88 157)(89 99)(90 159)(92 141)(94 143)(96 145)(98 147)(100 149)(121 131)(123 133)(125 135)(127 137)(129 139)(142 152)(144 154)(146 156)(148 158)(150 160)
(1 99)(2 100)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(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)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 101)(77 102)(78 103)(79 104)(80 105)(81 160)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)
(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)
(1 103 129 78)(2 102 130 77)(3 101 131 76)(4 120 132 75)(5 119 133 74)(6 118 134 73)(7 117 135 72)(8 116 136 71)(9 115 137 70)(10 114 138 69)(11 113 139 68)(12 112 140 67)(13 111 121 66)(14 110 122 65)(15 109 123 64)(16 108 124 63)(17 107 125 62)(18 106 126 61)(19 105 127 80)(20 104 128 79)(21 154 51 95)(22 153 52 94)(23 152 53 93)(24 151 54 92)(25 150 55 91)(26 149 56 90)(27 148 57 89)(28 147 58 88)(29 146 59 87)(30 145 60 86)(31 144 41 85)(32 143 42 84)(33 142 43 83)(34 141 44 82)(35 160 45 81)(36 159 46 100)(37 158 47 99)(38 157 48 98)(39 156 49 97)(40 155 50 96)```

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 4 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 C5⋊D4 D20 C4○D20 D4⋊2D5 kernel C23.14D20 C10.10C42 C2×C4⋊Dic5 C2×C23.D5 C10×C22⋊C4 C2×C20 C22×C10 C2×C22⋊C4 C2×C10 C22×C4 C24 C2×C4 C23 C22 C22 # reps 1 3 1 2 1 2 2 2 10 4 2 8 8 8 8

Matrix representation of C23.14D20 in GL6(𝔽41)

 1 0 0 0 0 0 23 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1
,
 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 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 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 40 0 0 0 0 0 0 40
,
 10 0 0 0 0 0 38 37 0 0 0 0 0 0 14 16 0 0 0 0 36 9 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 6 28 0 0 0 0 9 35 0 0 0 0 0 0 35 23 0 0 0 0 27 6 0 0 0 0 0 0 9 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [1,23,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[10,38,0,0,0,0,0,37,0,0,0,0,0,0,14,36,0,0,0,0,16,9,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[6,9,0,0,0,0,28,35,0,0,0,0,0,0,35,27,0,0,0,0,23,6,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;`

C23.14D20 in GAP, Magma, Sage, TeX

`C_2^3._{14}D_{20}`
`% in TeX`

`G:=Group("C2^3.14D20");`
`// GroupNames label`

`G:=SmallGroup(320,580);`
`// by ID`

`G=gap.SmallGroup(320,580);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,387,12550]);`
`// Polycyclic`

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

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