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G = C2×C23.18D10order 320 = 26·5

Direct product of C2 and C23.18D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23.18D10, C24.58D10, (C2×D4).228D10, (C23×Dic5)⋊8C2, (C22×D4).10D5, (C2×C20).642C23, (C2×C10).291C24, C10.139(C22×D4), (C22×C4).269D10, (C22×C10).121D4, C23.67(C5⋊D4), C23.D557C22, (D4×C10).311C22, C10.D472C22, C105(C22.D4), (C23×C10).73C22, C22.305(C23×D5), C23.133(C22×D5), C22.77(D42D5), (C22×C20).437C22, (C22×C10).227C23, (C2×Dic5).291C23, (C22×Dic5)⋊48C22, (D4×C2×C10).21C2, (C2×C10).73(C2×D4), C56(C2×C22.D4), C10.103(C2×C4○D4), C2.67(C2×D42D5), (C2×C23.D5)⋊24C2, C2.12(C22×C5⋊D4), (C2×C10.D4)⋊47C2, (C2×C4).236(C22×D5), C22.108(C2×C5⋊D4), (C2×C10).175(C4○D4), SmallGroup(320,1468)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C23.18D10
C1C5C10C2×C10C2×Dic5C22×Dic5C23×Dic5 — C2×C23.18D10
C5C2×C10 — C2×C23.18D10
C1C23C22×D4

Generators and relations for C2×C23.18D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 958 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C5, C2×C4 [×2], C2×C4 [×26], D4 [×8], C23, C23 [×8], C23 [×10], C10, C10 [×6], C10 [×6], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C24 [×2], Dic5 [×8], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×22], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×Dic5 [×8], C2×Dic5 [×16], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×C10, C22×C10 [×8], C22×C10 [×10], C2×C22.D4, C10.D4 [×8], C23.D5 [×12], C22×Dic5 [×8], C22×Dic5 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C2×C10.D4 [×2], C23.18D10 [×8], C2×C23.D5, C2×C23.D5 [×2], C23×Dic5, D4×C2×C10, C2×C23.18D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C22.D4, D42D5 [×4], C2×C5⋊D4 [×6], C23×D5, C23.18D10 [×4], C2×D42D5 [×2], C22×C5⋊D4, C2×C23.18D10

Smallest permutation representation of C2×C23.18D10
On 160 points
Generators in S160
(1 27)(2 28)(3 29)(4 30)(5 26)(6 21)(7 22)(8 23)(9 24)(10 25)(11 39)(12 40)(13 36)(14 37)(15 38)(16 33)(17 34)(18 35)(19 31)(20 32)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 61)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 117)(92 118)(93 119)(94 120)(95 111)(96 112)(97 113)(98 114)(99 115)(100 116)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 131)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
(1 41)(2 47)(3 43)(4 49)(5 45)(6 50)(7 46)(8 42)(9 48)(10 44)(11 51)(12 57)(13 53)(14 59)(15 55)(16 56)(17 52)(18 58)(19 54)(20 60)(21 61)(22 67)(23 63)(24 69)(25 65)(26 66)(27 62)(28 68)(29 64)(30 70)(31 79)(32 75)(33 71)(34 77)(35 73)(36 78)(37 74)(38 80)(39 76)(40 72)(81 141)(82 135)(83 143)(84 137)(85 145)(86 139)(87 147)(88 131)(89 149)(90 133)(91 136)(92 144)(93 138)(94 146)(95 140)(96 148)(97 132)(98 150)(99 134)(100 142)(101 113)(102 130)(103 115)(104 122)(105 117)(106 124)(107 119)(108 126)(109 111)(110 128)(112 151)(114 153)(116 155)(118 157)(120 159)(121 154)(123 156)(125 158)(127 160)(129 152)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)(21 39)(22 40)(23 36)(24 37)(25 38)(26 33)(27 34)(28 35)(29 31)(30 32)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(49 60)(50 51)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 91)(89 92)(90 93)(101 106)(102 107)(103 108)(104 109)(105 110)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 121)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 37)(32 38)(33 39)(34 40)(35 36)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 99)(82 100)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(101 152)(102 153)(103 154)(104 155)(105 156)(106 157)(107 158)(108 159)(109 160)(110 151)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(131 148)(132 149)(133 150)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)
(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)
(1 137 12 149)(2 131 13 143)(3 135 14 147)(4 139 15 141)(5 133 11 145)(6 150 16 138)(7 144 17 132)(8 148 18 136)(9 142 19 140)(10 146 20 134)(21 153 33 107)(22 157 34 101)(23 151 35 105)(24 155 31 109)(25 159 32 103)(26 102 39 158)(27 106 40 152)(28 110 36 156)(29 104 37 160)(30 108 38 154)(41 92 57 97)(42 88 58 83)(43 100 59 95)(44 86 60 81)(45 98 51 93)(46 84 52 89)(47 96 53 91)(48 82 54 87)(49 94 55 99)(50 90 56 85)(61 130 71 125)(62 118 72 113)(63 128 73 123)(64 116 74 111)(65 126 75 121)(66 114 76 119)(67 124 77 129)(68 112 78 117)(69 122 79 127)(70 120 80 115)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C···4J4K4L4M4N5A5B10A···10N10O···10AD20A···20H
order12···2222222444···444445510···1010···1020···20
size11···12222444410···1020202020222···24···44···4

68 irreducible representations

dim11111122222224
type+++++++++++-
imageC1C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D4D42D5
kernelC2×C23.18D10C2×C10.D4C23.18D10C2×C23.D5C23×Dic5D4×C2×C10C22×C10C22×D4C2×C10C22×C4C2×D4C24C23C22
# reps128311428284168

Matrix representation of C2×C23.18D10 in GL5(𝔽41)

400000
01000
00100
00010
00001
,
400000
01000
00100
000126
000040
,
10000
040000
004000
000400
000040
,
10000
01000
00100
000400
000040
,
400000
010000
011400
00010
0002240
,
400000
0191200
042200
0003212
00079

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,26,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,10,11,0,0,0,0,4,0,0,0,0,0,1,22,0,0,0,0,40],[40,0,0,0,0,0,19,4,0,0,0,12,22,0,0,0,0,0,32,7,0,0,0,12,9] >;

C2×C23.18D10 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{18}D_{10}
% in TeX

G:=Group("C2xC2^3.18D10");
// GroupNames label

G:=SmallGroup(320,1468);
// by ID

G=gap.SmallGroup(320,1468);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,675,297,12550]);
// Polycyclic

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

׿
×
𝔽