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G = C24.3D10  order 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.D5⋊14C4, C23.15(C4×D5), C22.95(D4×D5), C2.1(C20⋊2D4), (C22×C4).24D10, C10.82(C4⋊D4), (C2×Dic5).150D4, C10.30(C4.4D4), C22.50(C4○D20), (C23×C10).27C22, C5⋊5(C24.C22), C23.277(C22×D5), C10.10C42⋊38C2, C10.11(C42⋊2C2), C10.45(C42⋊C2), C2.24(Dic5⋊4D4), C22.43(D4⋊2D5), (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

Derived series
C1 — C2×C10 — C24.3D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C23.D5 — C24.3D10
Lower central
C5 — C2×C10 — C24.3D10
Upper central
C1 — C23 — C2×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, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C24.C22, C4×Dic5, C10.D4, C23.D5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C4×Dic5, C2×C10.D4, C2×C23.D5, C10×C22⋊C4, C24.3D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4×D5, C5⋊D4, C22×D5, C24.C22, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, C23.11D10, C23.D10, Dic5⋊4D4, Dic5.5D4, C4×C5⋊D4, C23.18D10, C20⋊2D4, C24.3D10

Smallest permutation representation of C24.3D10
►On 160 points
Generators in S160
(2 99)(4 81)(6 83)(8 85)(10 87)(12 89)(14 91)(16 93)(18 95)(20 97)(21 126)(22 155)(23 128)(24 157)(25 130)(26 159)(27 132)(28 141)(29 134)(30 143)(31 136)(32 145)(33 138)(34 147)(35 140)(36 149)(37 122)(38 151)(39 124)(40 153)(42 79)(44 61)(46 63)(48 65)(50 67)(52 69)(54 71)(56 73)(58 75)(60 77)(101 129)(102 158)(103 131)(104 160)(105 133)(106 142)(107 135)(108 144)(109 137)(110 146)(111 139)(112 148)(113 121)(114 150)(115 123)(116 152)(117 125)(118 154)(119 127)(120 156)

(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)(21 154)(22 155)(23 156)(24 157)(25 158)(26 159)(27 160)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 149)(37 150)(38 151)(39 152)(40 153)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)

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

(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)

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

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×D5D4⋊2D5
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
 

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