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## G = C24.6D10order 320 = 26·5

### 6th non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — C24.6D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C10.D4 — C24.6D10
 Lower central C5 — C22×C10 — C24.6D10
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.6D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 614 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, 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, C23.Q8, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C10.D4, C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C24.6D10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C4⋊D4, C22⋊Q8, C422C2, Dic10, C5⋊D4, C22×D5, C23.Q8, C2×Dic10, C4○D20, D4×D5, D42D5, C2×C5⋊D4, Dic5.14D4, C23.D10, D10⋊D4, C20.48D4, C202D4, Dic5⋊D4, C24.6D10

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 D4 D4 Q8 D5 C4○D4 D10 D10 C5⋊D4 Dic10 C4○D20 D4×D5 D4⋊2D5 kernel C24.6D10 C10.10C42 C2×C10.D4 C2×C4⋊Dic5 C2×C23.D5 C10×C22⋊C4 C2×Dic5 C2×C20 C22×C10 C2×C22⋊C4 C2×C10 C22×C4 C24 C2×C4 C23 C22 C22 C22 # reps 1 1 2 1 2 1 4 2 2 2 6 4 2 8 8 8 4 4

Matrix representation of C24.6D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 24 40 0 0 0 0 0 0 1 0 0 0 0 0 12 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
,
 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 20 0 0 0 0 0 16 39 0 0 0 0 0 0 25 0 0 0 0 0 24 23 0 0 0 0 0 0 12 39 0 0 0 0 10 29
,
 26 36 0 0 0 0 37 15 0 0 0 0 0 0 40 24 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 29 1

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,12,0,0,0,0,0,40],[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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[20,16,0,0,0,0,0,39,0,0,0,0,0,0,25,24,0,0,0,0,0,23,0,0,0,0,0,0,12,10,0,0,0,0,39,29],[26,37,0,0,0,0,36,15,0,0,0,0,0,0,40,0,0,0,0,0,24,1,0,0,0,0,0,0,40,29,0,0,0,0,0,1] >;`

C24.6D10 in GAP, Magma, Sage, TeX

`C_2^4._6D_{10}`
`% in TeX`

`G:=Group("C2^4.6D10");`
`// GroupNames label`

`G:=SmallGroup(320,575);`
`// by ID`

`G=gap.SmallGroup(320,575);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,387,12550]);`
`// Polycyclic`

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

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