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G = C24.3D10order 320 = 26·5

3rd non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.3D10, C10.71(C4×D4), (C2×C20).247D4, C23.D514C4, C23.15(C4×D5), C22.95(D4×D5), C2.1(C202D4), (C22×C4).24D10, C10.82(C4⋊D4), (C2×Dic5).150D4, C10.30(C4.4D4), C22.50(C4○D20), (C23×C10).27C22, C55(C24.C22), C23.277(C22×D5), C10.10C4238C2, C10.11(C422C2), C10.45(C42⋊C2), C2.24(Dic54D4), C22.43(D42D5), (C22×C20).341C22, (C22×C10).319C23, C2.5(Dic5.5D4), C2.5(C23.D10), C10.71(C22.D4), C2.2(C23.18D10), (C22×Dic5).35C22, C2.13(C23.11D10), C2.7(C4×C5⋊D4), (C2×C4×Dic5)⋊21C2, (C2×C22⋊C4).6D5, C22.123(C2×C4×D5), (C2×C10.D4)⋊7C2, (C2×C10).314(C2×D4), C22.47(C2×C5⋊D4), (C2×C23.D5).6C2, (C2×C4).167(C5⋊D4), (C10×C22⋊C4).23C2, (C2×C10).140(C4○D4), (C22×C10).115(C2×C4), (C2×C10).206(C22×C4), (C2×Dic5).103(C2×C4), SmallGroup(320,571)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.3D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C23.D5 — C24.3D10
C5C2×C10 — C24.3D10
C1C23C2×C22⋊C4

Generators and relations for C24.3D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, 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: 590 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C5, C2×C4 [×2], C2×C4 [×20], C23, C23 [×2], C23 [×6], C10 [×7], C10 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×7], C20 [×3], C2×C10 [×7], C2×C10 [×10], C2.C42 [×2], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×2], C2×C20 [×5], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.C22, C4×Dic5 [×2], C10.D4 [×2], C23.D5 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×2], C2×C4×Dic5, C2×C10.D4, C2×C23.D5 [×2], C10×C22⋊C4, C24.3D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22×C4, C2×D4 [×2], C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D42D5 [×3], C2×C5⋊D4, C23.11D10, C23.D10, Dic54D4, Dic5.5D4, C4×C5⋊D4, C23.18D10, C202D4, C24.3D10

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B10A···10N10O···10V20A···20P
order12···2224444444···444445510···1010···1020···20
size11···14422224410···1020202020222···24···44···4

68 irreducible representations

dim111111122222222244
type++++++++++++-
imageC1C2C2C2C2C2C4D4D4D5C4○D4D10D10C5⋊D4C4×D5C4○D20D4×D5D42D5
kernelC24.3D10C10.10C42C2×C4×Dic5C2×C10.D4C2×C23.D5C10×C22⋊C4C23.D5C2×Dic5C2×C20C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22C22
# reps121121822284288826

Matrix representation of C24.3D10 in GL6(𝔽41)

100000
15400000
001000
0004000
000010
00003640
,
100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
001000
000100
0000400
0000040
,
100000
010000
008000
000500
00004016
000001
,
40220000
010000
000500
008000
00003221
000009

G:=sub<GL(6,GF(41))| [1,15,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,36,0,0,0,0,0,40],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,40,0,0,0,0,0,16,1],[40,0,0,0,0,0,22,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,32,0,0,0,0,0,21,9] >;

C24.3D10 in GAP, Magma, Sage, TeX

C_2^4._3D_{10}
% in TeX

G:=Group("C2^4.3D10");
// GroupNames label

G:=SmallGroup(320,571);
// by ID

G=gap.SmallGroup(320,571);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,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