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## G = C42.155D10order 320 = 26·5

### 155th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.155D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊2Q8 — C42.155D10
 Lower central C5 — C2×C10 — C42.155D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.155D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2b2c9 >

Subgroups: 590 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4, C2×C4 [×6], C2×C4 [×9], Q8 [×4], C23, D5, C10, C10 [×2], C42, C42 [×5], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×14], C22×C4, C2×Q8 [×2], Dic5 [×7], C20 [×2], C20 [×6], D10 [×3], C2×C10, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C422C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C22×D5, C22.35C24, C4×Dic5, C4×Dic5 [×4], C10.D4 [×6], C4⋊Dic5 [×8], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×2], C2×C4×D5, C202Q8, C42⋊D5, Dic53Q8 [×2], C4.Dic10 [×4], D102Q8 [×2], C4⋊C4⋊D5 [×4], C5×C42.C2, C42.155D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2- 1+4 [×2], C22×D5 [×7], C22.35C24, Q82D5 [×2], C23×D5, C2×Q82D5, D4.10D10 [×2], C42.155D10

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 2- 1+4 Q8⋊2D5 D4.10D10 kernel C42.155D10 C20⋊2Q8 C42⋊D5 Dic5⋊3Q8 C4.Dic10 D10⋊2Q8 C4⋊C4⋊D5 C5×C42.C2 C42.C2 C20 C42 C4⋊C4 C10 C4 C2 # reps 1 1 1 2 4 2 4 1 2 4 2 12 2 4 8

Matrix representation of C42.155D10 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 11 13 0 0 0 0 0 0 19 30 0 0 0 0 11 13 0 0 0 0 0 0 19 30 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 22 19 32 9 0 0 0 0 22 12 32 10 0 0 0 0 32 9 19 22 0 0 0 0 32 10 19 29 0 0 0 0 0 0 0 0 7 8 21 18 0 0 0 0 18 25 20 0 0 0 0 0 20 23 34 33 0 0 0 0 21 0 23 16
,
 19 22 9 32 0 0 0 0 12 22 10 32 0 0 0 0 9 32 22 19 0 0 0 0 10 32 29 19 0 0 0 0 0 0 0 0 34 1 21 3 0 0 0 0 23 7 20 20 0 0 0 0 21 3 34 1 0 0 0 0 20 20 23 7

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0,0,0,11,19,0,0,0,0,0,0,13,30,0,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[22,22,32,32,0,0,0,0,19,12,9,10,0,0,0,0,32,32,19,19,0,0,0,0,9,10,22,29,0,0,0,0,0,0,0,0,7,18,20,21,0,0,0,0,8,25,23,0,0,0,0,0,21,20,34,23,0,0,0,0,18,0,33,16],[19,12,9,10,0,0,0,0,22,22,32,32,0,0,0,0,9,10,22,29,0,0,0,0,32,32,19,19,0,0,0,0,0,0,0,0,34,23,21,20,0,0,0,0,1,7,3,20,0,0,0,0,21,20,34,23,0,0,0,0,3,20,1,7] >;`

C42.155D10 in GAP, Magma, Sage, TeX

`C_4^2._{155}D_{10}`
`% in TeX`

`G:=Group("C4^2.155D10");`
`// GroupNames label`

`G:=SmallGroup(320,1369);`
`// by ID`

`G=gap.SmallGroup(320,1369);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,297,192,12550]);`
`// Polycyclic`

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

׿
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