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## G = C42.234D10order 320 = 26·5

### 54th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.234D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.234D10
 Lower central C5 — C2×C10 — C42.234D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.234D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 734 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×2], C42, C42 [×5], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C4×D5 [×6], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10 [×2], C23.36C23, C4×Dic5 [×3], C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×3], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C20.6Q8, D5×C42, C23.11D10 [×2], C23.D10 [×2], Dic54D4 [×2], D10.12D4 [×2], D4×Dic5, C202D4, Q8×Dic5, D103Q8, C5×C4.4D4, C42.234D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, D42D5 [×2], C23×D5, C2×D42D5, D5×C4○D4 [×2], C42.234D10

Smallest permutation representation of C42.234D10
On 160 points
Generators in S160
```(1 135 33 104)(2 114 34 88)(3 137 35 106)(4 116 36 90)(5 139 37 108)(6 118 38 82)(7 131 39 110)(8 120 40 84)(9 133 31 102)(10 112 32 86)(11 115 47 89)(12 138 48 107)(13 117 49 81)(14 140 50 109)(15 119 41 83)(16 132 42 101)(17 111 43 85)(18 134 44 103)(19 113 45 87)(20 136 46 105)(21 95 143 55)(22 79 144 64)(23 97 145 57)(24 71 146 66)(25 99 147 59)(26 73 148 68)(27 91 149 51)(28 75 150 70)(29 93 141 53)(30 77 142 62)(52 155 92 122)(54 157 94 124)(56 159 96 126)(58 151 98 128)(60 153 100 130)(61 156 76 123)(63 158 78 125)(65 160 80 127)(67 152 72 129)(69 154 74 121)
(1 100 19 73)(2 51 20 69)(3 92 11 75)(4 53 12 61)(5 94 13 77)(6 55 14 63)(7 96 15 79)(8 57 16 65)(9 98 17 71)(10 59 18 67)(21 140 158 118)(22 110 159 83)(23 132 160 120)(24 102 151 85)(25 134 152 112)(26 104 153 87)(27 136 154 114)(28 106 155 89)(29 138 156 116)(30 108 157 81)(31 58 43 66)(32 99 44 72)(33 60 45 68)(34 91 46 74)(35 52 47 70)(36 93 48 76)(37 54 49 62)(38 95 50 78)(39 56 41 64)(40 97 42 80)(82 143 109 125)(84 145 101 127)(86 147 103 129)(88 149 105 121)(90 141 107 123)(111 146 133 128)(113 148 135 130)(115 150 137 122)(117 142 139 124)(119 144 131 126)
(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 33 32)(2 31 34 9)(3 8 35 40)(4 39 36 7)(5 6 37 38)(11 16 47 42)(12 41 48 15)(13 14 49 50)(17 20 43 46)(18 45 44 19)(21 142 143 30)(22 29 144 141)(23 150 145 28)(24 27 146 149)(25 148 147 26)(51 98 91 58)(52 57 92 97)(53 96 93 56)(54 55 94 95)(59 100 99 60)(61 79 76 64)(62 63 77 78)(65 75 80 70)(66 69 71 74)(67 73 72 68)(81 82 117 118)(83 90 119 116)(84 115 120 89)(85 88 111 114)(86 113 112 87)(101 137 132 106)(102 105 133 136)(103 135 134 104)(107 131 138 110)(108 109 139 140)(121 151 154 128)(122 127 155 160)(123 159 156 126)(124 125 157 158)(129 153 152 130)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 C4○D4 D10 D10 D10 D10 D4⋊2D5 D5×C4○D4 kernel C42.234D10 C20.6Q8 D5×C42 C23.11D10 C23.D10 Dic5⋊4D4 D10.12D4 D4×Dic5 C20⋊2D4 Q8×Dic5 D10⋊3Q8 C5×C4.4D4 C4.4D4 Dic5 C20 D10 C42 C22⋊C4 C2×D4 C2×Q8 C4 C2 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 4 4 4 2 8 2 2 4 8

Matrix representation of C42.234D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 0 32 9
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 0 32 9
,
 6 6 0 0 0 0 35 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 39 0 0 0 0 0 40
,
 6 6 0 0 0 0 1 35 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 2 0 0 0 0 40 1

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

C42.234D10 in GAP, Magma, Sage, TeX

`C_4^2._{234}D_{10}`
`% in TeX`

`G:=Group("C4^2.234D10");`
`// GroupNames label`

`G:=SmallGroup(320,1352);`
`// by ID`

`G=gap.SmallGroup(320,1352);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,346,297,12550]);`
`// Polycyclic`

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

׿
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