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## G = C42.118D10order 320 = 26·5

### 118th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.118D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C4×C5⋊D4 — C42.118D10
 Lower central C5 — C2×C10 — C42.118D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.118D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 718 in 218 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23, D5, C10 [×3], C10 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×13], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×7], C20 [×5], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C422C2 [×2], Dic10, C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C22.33C24, C4×Dic5, C10.D4 [×10], C4⋊Dic5 [×3], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C20.6Q8, C422D5, Dic5.14D4, C23.D10, D10.12D4, C22.D20, Dic5.Q8, D10⋊Q8, C2×C10.D4, C20.48D4, C4×C5⋊D4, C23.23D10, C23.18D10, Dic5⋊D4, D4×C20, C42.118D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.33C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D4.10D10, C42.118D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H ··· 4N 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 20 2 2 2 2 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 2- 1+4 D4⋊6D10 D4.10D10 kernel C42.118D10 C20.6Q8 C42⋊2D5 Dic5.14D4 C23.D10 D10.12D4 C22.D20 Dic5.Q8 D10⋊Q8 C2×C10.D4 C20.48D4 C4×C5⋊D4 C23.23D10 C23.18D10 Dic5⋊D4 D4×C20 C4×D4 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C10 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 4 2 4 2 4 2 16 1 1 4 4

Matrix representation of C42.118D10 in GL6(𝔽41)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 16 18 33 0 0 17 34 0 33 0 0 0 25 18 25 0 0 24 30 28 31
,
 0 32 0 0 0 0 32 0 0 0 0 0 0 0 2 0 37 13 0 0 28 30 15 28 0 0 15 15 11 0 0 0 20 33 4 39
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 26 28 4 0 0 15 15 37 0 0 0 26 0 8 15 0 0 30 26 16 7
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 24 16 13 17 0 0 19 34 4 4 0 0 18 25 33 7 0 0 14 30 25 32

`G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,17,0,24,0,0,16,34,25,30,0,0,18,0,18,28,0,0,33,33,25,31],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,2,28,15,20,0,0,0,30,15,33,0,0,37,15,11,4,0,0,13,28,0,39],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,15,26,30,0,0,26,15,0,26,0,0,28,37,8,16,0,0,4,0,15,7],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,24,19,18,14,0,0,16,34,25,30,0,0,13,4,33,25,0,0,17,4,7,32] >;`

C42.118D10 in GAP, Magma, Sage, TeX

`C_4^2._{118}D_{10}`
`% in TeX`

`G:=Group("C4^2.118D10");`
`// GroupNames label`

`G:=SmallGroup(320,1236);`
`// by ID`

`G=gap.SmallGroup(320,1236);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,570,297,12550]);`
`// 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=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

׿
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