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## G = C42.145D10order 320 = 26·5

### 145th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.145D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C42.145D10
 Lower central C5 — C2×C10 — C42.145D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.145D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 854 in 220 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×6], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×6], C20 [×5], D10 [×6], C2×C10, C2×C10 [×6], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C42.C2, Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C5×D4, C5×Q8, C22×D5 [×2], C22×C10 [×2], C22.56C24, C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C20.6Q8, C4.D20, Dic5.14D4 [×2], D10.12D4 [×2], D10⋊D4 [×2], C22.D20 [×2], Dic5⋊D4 [×2], D103Q8 [×2], C5×C4.4D4, C42.145D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×2], 2- 1+4, C22×D5 [×7], C22.56C24, C23×D5, D46D10, D48D10, D4.10D10, C42.145D10

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊6D10 D4⋊8D10 D4.10D10 kernel C42.145D10 C20.6Q8 C4.D20 Dic5.14D4 D10.12D4 D10⋊D4 C22.D20 Dic5⋊D4 D10⋊3Q8 C5×C4.4D4 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C10 C10 C2 C2 C2 # reps 1 1 1 2 2 2 2 2 2 1 2 2 8 2 2 2 1 4 4 4

Matrix representation of C42.145D10 in GL8(𝔽41)

 39 9 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 0 11 9 0 0 0 0 0 0 32 30 0 0 0 0 0 0 0 0 12 10 23 7 0 0 0 0 9 22 9 14 0 0 0 0 0 16 10 9 0 0 0 0 25 16 32 38
,
 40 0 0 28 0 0 0 0 0 40 13 13 0 0 0 0 3 3 1 0 0 0 0 0 38 0 0 1 0 0 0 0 0 0 0 0 23 36 15 26 0 0 0 0 40 18 0 26 0 0 0 0 0 0 17 1 0 0 0 0 0 0 40 24
,
 5 22 30 35 0 0 0 0 28 33 22 36 0 0 0 0 0 28 27 19 0 0 0 0 17 13 22 17 0 0 0 0 0 0 0 0 20 25 25 38 0 0 0 0 5 33 18 7 0 0 0 0 32 37 3 34 0 0 0 0 32 0 36 26
,
 21 35 6 11 0 0 0 0 12 20 5 19 0 0 0 0 24 12 22 14 0 0 0 0 39 27 24 19 0 0 0 0 0 0 0 0 37 25 7 30 0 0 0 0 3 33 14 2 0 0 0 0 13 37 38 16 0 0 0 0 9 0 5 15

`G:=sub<GL(8,GF(41))| [39,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,12,9,0,25,0,0,0,0,10,22,16,16,0,0,0,0,23,9,10,32,0,0,0,0,7,14,9,38],[40,0,3,38,0,0,0,0,0,40,3,0,0,0,0,0,0,13,1,0,0,0,0,0,28,13,0,1,0,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18,0,0,0,0,0,0,15,0,17,40,0,0,0,0,26,26,1,24],[5,28,0,17,0,0,0,0,22,33,28,13,0,0,0,0,30,22,27,22,0,0,0,0,35,36,19,17,0,0,0,0,0,0,0,0,20,5,32,32,0,0,0,0,25,33,37,0,0,0,0,0,25,18,3,36,0,0,0,0,38,7,34,26],[21,12,24,39,0,0,0,0,35,20,12,27,0,0,0,0,6,5,22,24,0,0,0,0,11,19,14,19,0,0,0,0,0,0,0,0,37,3,13,9,0,0,0,0,25,33,37,0,0,0,0,0,7,14,38,5,0,0,0,0,30,2,16,15] >;`

C42.145D10 in GAP, Magma, Sage, TeX

`C_4^2._{145}D_{10}`
`% in TeX`

`G:=Group("C4^2.145D10");`
`// GroupNames label`

`G:=SmallGroup(320,1356);`
`// by ID`

`G=gap.SmallGroup(320,1356);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,136,12550]);`
`// Polycyclic`

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

׿
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