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## G = C42.152D10order 320 = 26·5

### 152nd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.152D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.152D10
 Lower central C5 — C2×C10 — C42.152D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.152D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 710 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×6], C20 [×2], C20 [×6], 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 [×2], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, C4×Dic10, C4×D20, C4.Dic10 [×2], D5×C4⋊C4, C4⋊C47D5, C4⋊C47D5 [×2], D10.13D4 [×2], D102Q8 [×2], C4⋊C4⋊D5 [×2], C5×C42.C2, C42.152D10
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, Q82D5 [×2], C23×D5, C2×Q82D5, D5×C4○D4, D4.10D10, C42.152D10

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

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

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

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

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 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 2- 1+4 Q8⋊2D5 D5×C4○D4 D4.10D10 kernel C42.152D10 C4×Dic10 C4×D20 C4.Dic10 D5×C4⋊C4 C4⋊C4⋊7D5 D10.13D4 D10⋊2Q8 C4⋊C4⋊D5 C5×C42.C2 C42.C2 C20 D10 C42 C4⋊C4 C10 C4 C2 C2 # reps 1 1 1 2 1 3 2 2 2 1 2 4 4 2 12 1 4 4 4

Matrix representation of C42.152D10 in GL6(𝔽41)

 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 9 0 0 0 0 0 1 32
,
 1 39 0 0 0 0 1 40 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 9 1
,
 9 23 0 0 0 0 0 32 0 0 0 0 0 0 0 6 0 0 0 0 34 7 0 0 0 0 0 0 32 39 0 0 0 0 40 9
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 7 35 0 0 0 0 8 34 0 0 0 0 0 0 9 2 0 0 0 0 1 32

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[9,0,0,0,0,0,23,32,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,32,40,0,0,0,0,39,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,7,8,0,0,0,0,35,34,0,0,0,0,0,0,9,1,0,0,0,0,2,32] >;`

C42.152D10 in GAP, Magma, Sage, TeX

`C_4^2._{152}D_{10}`
`% in TeX`

`G:=Group("C4^2.152D10");`
`// GroupNames label`

`G:=SmallGroup(320,1366);`
`// by ID`

`G=gap.SmallGroup(320,1366);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,297,192,12550]);`
`// 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^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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