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G = C23.14D20order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.14D20, C24.10D10, (C2×C20).51D4, C2.6(C207D4), (C22×C10).65D4, (C22×C4).91D10, C10.58(C4⋊D4), C22.125(C2×D20), C54(C23.11D4), C10.38(C4.4D4), C2.6(C20.17D4), C22.98(C4○D20), (C23×C10).36C22, (C22×C20).60C22, C23.370(C22×D5), C10.10C4215C2, C10.16(C422C2), C22.96(D42D5), (C22×C10).328C23, C10.73(C22.D4), C2.16(C22.D20), C2.8(C23.18D10), C2.14(C23.D10), (C22×Dic5).42C22, (C2×C4⋊Dic5)⋊12C2, (C2×C10).432(C2×D4), (C2×C4).30(C5⋊D4), (C2×C22⋊C4).15D5, (C10×C22⋊C4).16C2, C22.126(C2×C5⋊D4), (C2×C23.D5).15C2, (C2×C10).144(C4○D4), SmallGroup(320,580)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C23.14D20
C1C5C10C2×C10C22×C10C22×Dic5C2×C4⋊Dic5 — C23.14D20
C5C22×C10 — C23.14D20
C1C23C2×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 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×17], C23, C23 [×2], C23 [×6], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×4], C20 [×3], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×Dic5 [×12], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C23.11D4, C4⋊Dic5 [×2], C23.D5 [×4], C5×C22⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42, C10.10C42 [×2], C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C23.14D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4 [×5], D10 [×3], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.11D4, C2×D20, C4○D20, D42D5 [×4], C2×C5⋊D4, C23.D10 [×2], C22.D20 [×2], C207D4, C23.18D10, C20.17D4, C23.14D20

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

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12···22244444···45510···1010···1020···20
size11···144444420···20222···24···44···4

62 irreducible representations

dim111112222222224
type+++++++++++-
imageC1C2C2C2C2D4D4D5C4○D4D10D10C5⋊D4D20C4○D20D42D5
kernelC23.14D20C10.10C42C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C2×C20C22×C10C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22
# reps1312122210428888

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

100000
23400000
0040000
0004000
0000400
000001
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
001000
000100
0000400
0000040
,
1000000
38370000
00141600
0036900
000001
0000400
,
6280000
9350000
00352300
0027600
000090
0000032

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

׿
×
𝔽