Copied to
clipboard

## G = C42.161D10order 320 = 26·5

### 161st non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.161D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.161D10
 Lower central C5 — C2×C10 — C42.161D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C42.161D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 798 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], D5 [×3], C10 [×3], C10, C42, C42, C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×11], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×6], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C422C2, C422C2, Dic10, C4×D5 [×5], D20 [×2], C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×C10, C22.33C24, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×5], D10⋊C4 [×6], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], C20.6Q8, C4×D20, Dic5.14D4, Dic54D4, D10.12D4 [×2], D10⋊D4, C22.D20, C4.Dic10, D5×C4⋊C4, D10.13D4, D10⋊Q8, D102Q8, C4⋊C4⋊D5, C5×C422C2, C42.161D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.33C24, C23×D5, D5×C4○D4, D48D10, D4.10D10, C42.161D10

Smallest permutation representation of C42.161D10
On 160 points
Generators in S160
```(1 113 77 147)(2 104 78 158)(3 115 79 149)(4 106 80 160)(5 117 61 151)(6 108 62 142)(7 119 63 153)(8 110 64 144)(9 101 65 155)(10 112 66 146)(11 103 67 157)(12 114 68 148)(13 105 69 159)(14 116 70 150)(15 107 71 141)(16 118 72 152)(17 109 73 143)(18 120 74 154)(19 111 75 145)(20 102 76 156)(21 84 41 126)(22 95 42 137)(23 86 43 128)(24 97 44 139)(25 88 45 130)(26 99 46 121)(27 90 47 132)(28 81 48 123)(29 92 49 134)(30 83 50 125)(31 94 51 136)(32 85 52 127)(33 96 53 138)(34 87 54 129)(35 98 55 140)(36 89 56 131)(37 100 57 122)(38 91 58 133)(39 82 59 124)(40 93 60 135)
(1 43 11 53)(2 34 12 24)(3 45 13 55)(4 36 14 26)(5 47 15 57)(6 38 16 28)(7 49 17 59)(8 40 18 30)(9 51 19 41)(10 22 20 32)(21 65 31 75)(23 67 33 77)(25 69 35 79)(27 71 37 61)(29 73 39 63)(42 76 52 66)(44 78 54 68)(46 80 56 70)(48 62 58 72)(50 64 60 74)(81 108 91 118)(82 153 92 143)(83 110 93 120)(84 155 94 145)(85 112 95 102)(86 157 96 147)(87 114 97 104)(88 159 98 149)(89 116 99 106)(90 141 100 151)(101 136 111 126)(103 138 113 128)(105 140 115 130)(107 122 117 132)(109 124 119 134)(121 160 131 150)(123 142 133 152)(125 144 135 154)(127 146 137 156)(129 148 139 158)
(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 42 11 52)(2 51 12 41)(3 60 13 50)(4 49 14 59)(5 58 15 48)(6 47 16 57)(7 56 17 46)(8 45 18 55)(9 54 19 44)(10 43 20 53)(21 78 31 68)(22 67 32 77)(23 76 33 66)(24 65 34 75)(25 74 35 64)(26 63 36 73)(27 72 37 62)(28 61 38 71)(29 70 39 80)(30 79 40 69)(81 141 91 151)(82 150 92 160)(83 159 93 149)(84 148 94 158)(85 157 95 147)(86 146 96 156)(87 155 97 145)(88 144 98 154)(89 153 99 143)(90 142 100 152)(101 139 111 129)(102 128 112 138)(103 137 113 127)(104 126 114 136)(105 135 115 125)(106 124 116 134)(107 133 117 123)(108 122 118 132)(109 131 119 121)(110 140 120 130)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 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 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 2- 1+4 D5×C4○D4 D4⋊8D10 D4.10D10 kernel C42.161D10 C20.6Q8 C4×D20 Dic5.14D4 Dic5⋊4D4 D10.12D4 D10⋊D4 C22.D20 C4.Dic10 D5×C4⋊C4 D10.13D4 D10⋊Q8 D10⋊2Q8 C4⋊C4⋊D5 C5×C42⋊2C2 C42⋊2C2 D10 C42 C22⋊C4 C4⋊C4 C10 C10 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 4 2 6 6 1 1 4 4 4

Matrix representation of C42.161D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 39 0 0 0 0 1 0 39 0 0 1 0 40 0 0 0 0 1 0 40
,
 1 0 0 0 0 0 8 40 0 0 0 0 0 0 30 32 0 0 0 0 9 11 0 0 0 0 0 0 30 32 0 0 0 0 9 11
,
 33 2 0 0 0 0 30 8 0 0 0 0 0 0 15 15 13 13 0 0 26 33 28 4 0 0 1 1 26 26 0 0 40 35 15 8
,
 33 2 0 0 0 0 30 8 0 0 0 0 0 0 11 30 15 26 0 0 35 30 3 26 0 0 39 2 30 11 0 0 16 2 6 11

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,39,0,40,0,0,0,0,39,0,40],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[33,30,0,0,0,0,2,8,0,0,0,0,0,0,15,26,1,40,0,0,15,33,1,35,0,0,13,28,26,15,0,0,13,4,26,8],[33,30,0,0,0,0,2,8,0,0,0,0,0,0,11,35,39,16,0,0,30,30,2,2,0,0,15,3,30,6,0,0,26,26,11,11] >;`

C42.161D10 in GAP, Magma, Sage, TeX

`C_4^2._{161}D_{10}`
`% in TeX`

`G:=Group("C4^2.161D10");`
`// GroupNames label`

`G:=SmallGroup(320,1379);`
`// by ID`

`G=gap.SmallGroup(320,1379);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,80,12550]);`
`// 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=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
×
𝔽