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G = C24.19D10order 320 = 26·5

19th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.19D10, (D4×C10)⋊19C4, (C2×D4)⋊7Dic5, C206(C22⋊C4), (C2×Dic5)⋊12D4, C10.132(C4×D4), (C2×C20).191D4, C41(C23.D5), C2.19(D4×Dic5), (C22×D4).7D5, C2.4(C20⋊D4), C2.5(C202D4), C22.121(D4×D5), C10.36(C41D4), (C22×C4).353D10, C10.129(C4⋊D4), C23.10(C2×Dic5), C2.4(C20.17D4), C10.47(C4.4D4), (C23×C10).47C22, C55(C24.3C22), C23.306(C22×D5), C22.62(D42D5), (C22×C20).199C22, (C22×C10).366C23, C22.52(C22×Dic5), (C22×Dic5).220C22, (C2×C4×Dic5)⋊3C2, (D4×C2×C10).5C2, (C2×C4⋊Dic5)⋊35C2, (C2×C20).291(C2×C4), (C2×C10).555(C2×D4), (C2×C23.D5)⋊10C2, (C2×C4).50(C2×Dic5), C22.92(C2×C5⋊D4), C2.12(C2×C23.D5), (C2×C4).148(C5⋊D4), C10.117(C2×C22⋊C4), (C2×C10).162(C4○D4), (C22×C10).141(C2×C4), (C2×C10).298(C22×C4), SmallGroup(320,848)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.19D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C4×Dic5 — C24.19D10
C5C2×C10 — C24.19D10
C1C23C22×D4

Generators and relations for C24.19D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 766 in 258 conjugacy classes, 91 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C5, C2×C4 [×6], C2×C4 [×14], D4 [×8], C23, C23 [×4], C23 [×12], C10 [×3], C10 [×4], C10 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], Dic5 [×6], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2×C10 [×20], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×6], C5×D4 [×8], C22×C10, C22×C10 [×4], C22×C10 [×12], C24.3C22, C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×8], C22×Dic5 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C2×C4×Dic5, C2×C4⋊Dic5, C2×C23.D5 [×4], D4×C2×C10, C24.19D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C24.3C22, C23.D5 [×4], D4×D5 [×2], D42D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], D4×Dic5 [×2], C20.17D4, C202D4 [×2], C20⋊D4, C2×C23.D5, C24.19D10

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M4N4O4P5A5B10A···10N10O···10AD20A···20H
order12···2222244444···444445510···1010···1020···20
size11···14444222210···1020202020222···24···44···4

68 irreducible representations

dim1111112222222244
type+++++++++-++-
imageC1C2C2C2C2C4D4D4D5C4○D4D10Dic5D10C5⋊D4D4×D5D42D5
kernelC24.19D10C2×C4×Dic5C2×C4⋊Dic5C2×C23.D5D4×C2×C10D4×C10C2×Dic5C2×C20C22×D4C2×C10C22×C4C2×D4C24C2×C4C22C22
# reps11141844242841644

Matrix representation of C24.19D10 in GL6(𝔽41)

100000
010000
000100
001000
000010
0000940
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
0040000
0004000
000010
000001
,
670000
3500000
000100
0040000
000010
000001
,
2180000
2390000
009000
000900
0000118
0000940

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,35,0,0,0,0,7,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,2,0,0,0,0,18,39,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,9,0,0,0,0,18,40] >;

C24.19D10 in GAP, Magma, Sage, TeX

C_2^4._{19}D_{10}
% in TeX

G:=Group("C2^4.19D10");
// GroupNames label

G:=SmallGroup(320,848);
// by ID

G=gap.SmallGroup(320,848);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,12550]);
// Polycyclic

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

׿
×
𝔽