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## G = C24.13D10order 320 = 26·5

### 13rd non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.13D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×C5⋊D4 — C24.13D10
 Lower central C5 — C2×C10 — C24.13D10
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.13D10
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=de9 >

Subgroups: 1022 in 258 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C24.3C22, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C2×C4×Dic5, C2×C10.D4, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C24.13D10
Quotients:

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 20 20 2 2 2 2 4 4 10 ··· 10 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 D5 C4○D4 D10 D10 C5⋊D4 C4×D5 C4○D20 D4×D5 D4⋊2D5 kernel C24.13D10 C2×C4×Dic5 C2×C10.D4 C2×D10⋊C4 C2×C23.D5 C10×C22⋊C4 C22×C5⋊D4 C2×C5⋊D4 C2×Dic5 C2×C20 C2×C22⋊C4 C2×C10 C22×C4 C24 C2×C4 C23 C22 C22 C22 # reps 1 1 1 2 1 1 1 8 6 2 2 4 4 2 8 8 8 6 2

Matrix representation of C24.13D10 in GL5(𝔽41)

 40 0 0 0 0 0 23 1 0 0 0 5 18 0 0 0 0 0 40 0 0 0 0 40 1
,
 40 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 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 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 22 0 0 0 28 13 0 0 0 0 0 32 18 0 0 0 0 9
,
 9 0 0 0 0 0 28 22 0 0 0 37 13 0 0 0 0 0 9 23 0 0 0 9 32

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,5,0,0,0,1,18,0,0,0,0,0,40,40,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,0,28,0,0,0,22,13,0,0,0,0,0,32,0,0,0,0,18,9],[9,0,0,0,0,0,28,37,0,0,0,22,13,0,0,0,0,0,9,9,0,0,0,23,32] >;`

C24.13D10 in GAP, Magma, Sage, TeX

`C_2^4._{13}D_{10}`
`% in TeX`

`G:=Group("C2^4.13D10");`
`// GroupNames label`

`G:=SmallGroup(320,584);`
`// by ID`

`G=gap.SmallGroup(320,584);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,387,58,12550]);`
`// 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=d*e^9>;`
`// generators/relations`

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