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## G = C42.135D10order 320 = 26·5

### 135th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.135D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C4×D20 — C42.135D10
 Lower central C5 — C2×C10 — C42.135D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.135D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c9 >

Subgroups: 694 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C422C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.50C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C4×Dic10, C202Q8, C4×D20, C4.D20, C4⋊C47D5, C4⋊C4⋊D5, D103Q8, Q8×C20, C42.135D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, C4○D20, Q82D5, C23×D5, C2×C4○D20, C2×Q82D5, D4.10D10, C42.135D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 C4○D20 2- 1+4 Q8⋊2D5 D4.10D10 kernel C42.135D10 C4×Dic10 C20⋊2Q8 C4×D20 C4.D20 C4⋊C4⋊7D5 C4⋊C4⋊D5 D10⋊3Q8 Q8×C20 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C4 C10 C4 C2 # reps 1 2 1 1 2 2 4 2 1 2 8 6 6 2 16 1 4 4

Matrix representation of C42.135D10 in GL4(𝔽41) generated by

 2 32 0 0 37 39 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 40 23 0 0 32 1
,
 35 34 0 0 6 0 0 0 0 0 9 0 0 0 40 32
,
 6 40 0 0 35 35 0 0 0 0 9 0 0 0 0 9
`G:=sub<GL(4,GF(41))| [2,37,0,0,32,39,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,40,32,0,0,23,1],[35,6,0,0,34,0,0,0,0,0,9,40,0,0,0,32],[6,35,0,0,40,35,0,0,0,0,9,0,0,0,0,9] >;`

C42.135D10 in GAP, Magma, Sage, TeX

`C_4^2._{135}D_{10}`
`% in TeX`

`G:=Group("C4^2.135D10");`
`// GroupNames label`

`G:=SmallGroup(320,1256);`
`// by ID`

`G=gap.SmallGroup(320,1256);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,185,192,12550]);`
`// Polycyclic`

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

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