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## G = C42.151D10order 320 = 26·5

### 151st non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.151D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C42⋊D5 — C42.151D10
 Lower central C5 — C2×C10 — C42.151D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.151D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c9 >

Subgroups: 710 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×7], C5, C2×C4 [×7], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×7], D10 [×2], D10 [×5], C2×C10, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C2×C20 [×7], C22×D5 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20, C5×C4⋊C4 [×6], C2×Dic10, C2×C4×D5 [×4], C2×D20, C42⋊D5 [×2], Dic53Q8, Dic5.Q8 [×2], D5×C4⋊C4, C4⋊C47D5, D208C4, D10.13D4 [×2], D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.151D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.46C24, C23×D5, Q8.10D10, D5×C4○D4 [×2], C42.151D10

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

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 2- 1+4 Q8.10D10 D5×C4○D4 kernel C42.151D10 C42⋊D5 Dic5⋊3Q8 Dic5.Q8 D5×C4⋊C4 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 D10⋊Q8 C4⋊C4⋊D5 C5×C42.C2 C42.C2 Dic5 D10 C42 C4⋊C4 C10 C2 C2 # reps 1 2 1 2 1 1 1 2 2 2 1 2 4 4 2 12 1 4 8

Matrix representation of C42.151D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 32 0 0 0 0 9 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 34 0 0 0 0 7 34 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 40 0 0 0 0 0 34 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 40 0 0 0 0 40 0

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

C42.151D10 in GAP, Magma, Sage, TeX

`C_4^2._{151}D_{10}`
`% in TeX`

`G:=Group("C4^2.151D10");`
`// GroupNames label`

`G:=SmallGroup(320,1365);`
`// by ID`

`G=gap.SmallGroup(320,1365);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,346,297,136,12550]);`
`// Polycyclic`

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

׿
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