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## G = C42.134D10order 320 = 26·5

### 134th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.134D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊2Q8 — C42.134D10
 Lower central C5 — C2×C10 — C42.134D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.134D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 574 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], Q8 [×4], C23, D5, C10 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C2×Q8, C2×Q8, Dic5 [×7], C20 [×2], C20 [×6], D10 [×3], C2×C10, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8 [×2], C42.C2 [×5], C422C2 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22.35C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×10], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×2], D10⋊C4 [×4], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, Q8×C10, C4×Dic10, C20.6Q8 [×2], C42⋊D5, C422D5 [×2], Dic5.Q8 [×2], C4.Dic10, D102Q8, C4⋊C4⋊D5 [×2], Dic5⋊Q8, D103Q8, Q8×C20, C42.134D10
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, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D4.10D10, C42.134D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 4J 4K ··· 4Q 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 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 4 4 20 ··· 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 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 C2 C2 C2 D5 C4○D4 D10 D10 D10 C4○D20 2- 1+4 Q8.10D10 D4.10D10 kernel C42.134D10 C4×Dic10 C20.6Q8 C42⋊D5 C42⋊2D5 Dic5.Q8 C4.Dic10 D10⋊2Q8 C4⋊C4⋊D5 Dic5⋊Q8 D10⋊3Q8 Q8×C20 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C4 C10 C2 C2 # reps 1 1 2 1 2 2 1 1 2 1 1 1 2 4 6 6 2 16 2 4 4

Matrix representation of C42.134D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 0 14 11 0 0 0 31 14 14 0 0 21 4 10 0 0 0 20 21 0 10
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 17 40 0 0 0 0 1 24 0 0 0 0 0 0 24 40 0 0 0 0 1 17
,
 15 37 0 0 0 0 36 26 0 0 0 0 0 0 7 7 32 11 0 0 34 40 11 32 0 0 19 28 1 34 0 0 23 19 7 34
,
 26 4 0 0 0 0 26 15 0 0 0 0 0 0 7 7 32 11 0 0 40 34 11 14 0 0 28 19 1 0 0 0 19 23 7 40

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,0,21,20,0,0,0,31,4,21,0,0,14,14,10,0,0,0,11,14,0,10],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[15,36,0,0,0,0,37,26,0,0,0,0,0,0,7,34,19,23,0,0,7,40,28,19,0,0,32,11,1,7,0,0,11,32,34,34],[26,26,0,0,0,0,4,15,0,0,0,0,0,0,7,40,28,19,0,0,7,34,19,23,0,0,32,11,1,7,0,0,11,14,0,40] >;`

C42.134D10 in GAP, Magma, Sage, TeX

`C_4^2._{134}D_{10}`
`% in TeX`

`G:=Group("C4^2.134D10");`
`// GroupNames label`

`G:=SmallGroup(320,1255);`
`// by ID`

`G=gap.SmallGroup(320,1255);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,136,12550]);`
`// Polycyclic`

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

׿
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