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G = C24.62D10order 320 = 26·5

2nd non-split extension by C24 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.62D10, C23.15Dic10, (C23×C4).4D5, (C2×C20).451D4, C10.114(C4×D4), (C23×C20).2C2, C23.D517C4, C23.37(C4×D5), C10.69C22≀C2, (C22×C10).25Q8, C56(C23.8Q8), (C22×C4).406D10, (C22×C10).191D4, C2.1(C242D5), C23.81(C5⋊D4), C10.67(C22⋊Q8), C2.4(C20.48D4), C22.61(C4○D20), C223(C10.D4), (C23×C10).97C22, C22.31(C2×Dic10), C23.301(C22×D5), C10.10C4222C2, (C22×C20).482C22, (C22×C10).361C23, C10.67(C22.D4), C2.3(C23.23D10), (C22×Dic5).64C22, (C2×C10)⋊9(C4⋊C4), C10.78(C2×C4⋊C4), C2.28(C4×C5⋊D4), (C2×C10).43(C2×Q8), C22.149(C2×C4×D5), (C2×Dic5)⋊10(C2×C4), (C2×C10).547(C2×D4), C22.85(C2×C5⋊D4), (C2×C10).89(C4○D4), (C2×C10.D4)⋊15C2, (C2×C4).224(C5⋊D4), C2.20(C2×C10.D4), (C2×C23.D5).16C2, (C2×C10).243(C22×C4), (C22×C10).167(C2×C4), SmallGroup(320,837)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.62D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C23.D5 — C24.62D10
C5C2×C10 — C24.62D10
C1C23C23×C4

Generators and relations for C24.62D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de9 >

Subgroups: 638 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×4], C2×C4 [×26], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×10], C24, Dic5 [×6], C20 [×4], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×4], C2×C20 [×12], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.8Q8, C10.D4 [×4], C23.D5 [×4], C23.D5 [×2], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20 [×2], C22×C20 [×6], C23×C10, C10.10C42 [×2], C2×C10.D4 [×2], C2×C23.D5 [×2], C23×C20, C24.62D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×6], C22×D5, C23.8Q8, C10.D4 [×4], C2×Dic10, C2×C4×D5, C4○D20 [×2], C2×C5⋊D4 [×3], C2×C10.D4, C20.48D4 [×2], C4×C5⋊D4 [×2], C23.23D10, C242D5, C24.62D10

Smallest permutation representation of C24.62D10
On 160 points
Generators in S160
(1 136)(2 137)(3 138)(4 139)(5 140)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)
(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)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 60)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)(79 81)(80 82)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)
(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 25 11 35)(2 53 12 43)(3 23 13 33)(4 51 14 41)(5 21 15 31)(6 49 16 59)(7 39 17 29)(8 47 18 57)(9 37 19 27)(10 45 20 55)(22 139 32 129)(24 137 34 127)(26 135 36 125)(28 133 38 123)(30 131 40 121)(42 128 52 138)(44 126 54 136)(46 124 56 134)(48 122 58 132)(50 140 60 130)(61 150 71 160)(62 111 72 101)(63 148 73 158)(64 109 74 119)(65 146 75 156)(66 107 76 117)(67 144 77 154)(68 105 78 115)(69 142 79 152)(70 103 80 113)(81 104 91 114)(82 141 92 151)(83 102 93 112)(84 159 94 149)(85 120 95 110)(86 157 96 147)(87 118 97 108)(88 155 98 145)(89 116 99 106)(90 153 100 143)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B10A···10AD20A···20AF
order12···222224···44···45510···1020···20
size11···122222···220···20222···22···2

92 irreducible representations

dim111111222222222222
type+++++++-+++-
imageC1C2C2C2C2C4D4D4Q8D5C4○D4D10D10C5⋊D4Dic10C4×D5C5⋊D4C4○D20
kernelC24.62D10C10.10C42C2×C10.D4C2×C23.D5C23×C20C23.D5C2×C20C22×C10C22×C10C23×C4C2×C10C22×C4C24C2×C4C23C23C23C22
# reps12221842224421688816

Matrix representation of C24.62D10 in GL5(𝔽41)

400000
040000
00100
000400
000040
,
400000
040000
004000
00010
00001
,
10000
040000
004000
00010
00001
,
10000
040000
004000
000400
000040
,
320000
05000
003300
000100
00004
,
320000
00800
05000
000037
000100

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[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],[32,0,0,0,0,0,5,0,0,0,0,0,33,0,0,0,0,0,10,0,0,0,0,0,4],[32,0,0,0,0,0,0,5,0,0,0,8,0,0,0,0,0,0,0,10,0,0,0,37,0] >;

C24.62D10 in GAP, Magma, Sage, TeX

C_2^4._{62}D_{10}
% in TeX

G:=Group("C2^4.62D10");
// GroupNames label

G:=SmallGroup(320,837);
// by ID

G=gap.SmallGroup(320,837);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,758,58,12550]);
// Polycyclic

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

׿
×
𝔽