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## G = C42.240D10order 320 = 26·5

### 60th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.240D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.240D10
 Lower central C5 — C2×C10 — C42.240D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.240D10
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=b2c9 >

Subgroups: 1230 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C4 [×8], C22, C22 [×16], C5, C2×C4, C2×C4 [×6], C2×C4 [×19], D4 [×20], Q8 [×4], C23 [×5], D5 [×6], C10, C10 [×2], C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×6], C20 [×4], D10 [×2], D10 [×14], C2×C10, C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C4×D5 [×4], C4×D5 [×12], D20 [×20], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22×D5 [×4], C22.26C24, C4×Dic5, C4×Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5, C2×C4×D5 [×6], C2×D20 [×10], Q82D5 [×8], Q8×C10 [×2], D5×C42, C204D4, D208C4 [×4], C4⋊D20 [×4], C20.23D4 [×2], C5×C4⋊Q8, C2×Q82D5 [×2], C42.240D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], Q82D5 [×4], C23×D5, C2×D4×D5, C2×Q82D5 [×2], C42.240D10

Smallest permutation representation of C42.240D10
On 160 points
Generators in S160
```(1 100 21 61)(2 62 22 81)(3 82 23 63)(4 64 24 83)(5 84 25 65)(6 66 26 85)(7 86 27 67)(8 68 28 87)(9 88 29 69)(10 70 30 89)(11 90 31 71)(12 72 32 91)(13 92 33 73)(14 74 34 93)(15 94 35 75)(16 76 36 95)(17 96 37 77)(18 78 38 97)(19 98 39 79)(20 80 40 99)(41 108 146 123)(42 124 147 109)(43 110 148 125)(44 126 149 111)(45 112 150 127)(46 128 151 113)(47 114 152 129)(48 130 153 115)(49 116 154 131)(50 132 155 117)(51 118 156 133)(52 134 157 119)(53 120 158 135)(54 136 159 101)(55 102 160 137)(56 138 141 103)(57 104 142 139)(58 140 143 105)(59 106 144 121)(60 122 145 107)
(1 153 11 143)(2 144 12 154)(3 155 13 145)(4 146 14 156)(5 157 15 147)(6 148 16 158)(7 159 17 149)(8 150 18 160)(9 141 19 151)(10 152 20 142)(21 48 31 58)(22 59 32 49)(23 50 33 60)(24 41 34 51)(25 52 35 42)(26 43 36 53)(27 54 37 44)(28 45 38 55)(29 56 39 46)(30 47 40 57)(61 130 71 140)(62 121 72 131)(63 132 73 122)(64 123 74 133)(65 134 75 124)(66 125 76 135)(67 136 77 126)(68 127 78 137)(69 138 79 128)(70 129 80 139)(81 106 91 116)(82 117 92 107)(83 108 93 118)(84 119 94 109)(85 110 95 120)(86 101 96 111)(87 112 97 102)(88 103 98 113)(89 114 99 104)(90 105 100 115)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 84)(62 83)(63 82)(64 81)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 140)(120 139)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 20 20 20 20 2 ··· 2 4 4 4 4 5 5 5 5 10 10 10 10 2 2 2 ··· 2 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D4×D5 Q8⋊2D5 kernel C42.240D10 D5×C42 C20⋊4D4 D20⋊8C4 C4⋊D20 C20.23D4 C5×C4⋊Q8 C2×Q8⋊2D5 C4×D5 C4⋊Q8 C20 C42 C4⋊C4 C2×Q8 C4 C4 # reps 1 1 1 4 4 2 1 2 4 2 8 2 8 4 4 8

Matrix representation of C42.240D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 18 0 0 0 0 0 9 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 9 23 0 0 0 0 0 32 0 0 0 0 0 0 9 18 0 0 0 0 0 32
,
 34 34 0 0 0 0 7 1 0 0 0 0 0 0 1 39 0 0 0 0 1 40 0 0 0 0 0 0 40 39 0 0 0 0 1 1
,
 7 7 0 0 0 0 40 34 0 0 0 0 0 0 1 0 0 0 0 0 1 40 0 0 0 0 0 0 40 0 0 0 0 0 1 1

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,18,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,23,32,0,0,0,0,0,0,9,0,0,0,0,0,18,32],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,0,1] >;`

C42.240D10 in GAP, Magma, Sage, TeX

`C_4^2._{240}D_{10}`
`% in TeX`

`G:=Group("C4^2.240D10");`
`// GroupNames label`

`G:=SmallGroup(320,1397);`
`// by ID`

`G=gap.SmallGroup(320,1397);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,185,80,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^9>;`
`// generators/relations`

׿
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