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## G = C42.138D10order 320 = 26·5

### 138th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.138D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.11D10 — C42.138D10
 Lower central C5 — C2×C10 — C42.138D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.138D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 854 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×13], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×4], Dic5 [×4], C20 [×5], D10 [×6], C2×C10, C2×C10 [×6], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4, C4.4D4 [×3], C4⋊Q8, Dic10, C4×D5 [×4], D20, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4, C5×Q8, C22×D5 [×2], C22×C10 [×2], C22.49C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×6], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C42⋊D5 [×2], C23.11D10 [×2], Dic54D4 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], Dic5⋊D4 [×2], Dic5⋊Q8, C20.23D4, C5×C4.4D4, C42.138D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.49C24, C23×D5, D46D10, D5×C4○D4 [×2], C42.138D10

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

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 4P 4Q 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 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 20 20 2 2 2 2 4 4 4 10 ··· 10 20 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

53 irreducible representations

 dim 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 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D5×C4○D4 kernel C42.138D10 C42⋊D5 C23.11D10 Dic5⋊4D4 D10⋊D4 Dic5.5D4 Dic5⋊D4 Dic5⋊Q8 C20.23D4 C5×C4.4D4 C4.4D4 Dic5 C42 C22⋊C4 C2×D4 C2×Q8 C10 C2 C2 # reps 1 2 2 2 2 2 2 1 1 1 2 8 2 8 2 2 1 4 8

Matrix representation of C42.138D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 8 0 0 0 0 13 26
,
 1 0 0 0 0 0 21 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 9 5 0 0 0 0 25 32 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 27 40
,
 9 5 0 0 0 0 25 32 0 0 0 0 0 0 6 6 0 0 0 0 1 35 0 0 0 0 0 0 32 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,13,0,0,0,0,8,26],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,1,27,0,0,0,0,0,40],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;`

C42.138D10 in GAP, Magma, Sage, TeX

`C_4^2._{138}D_{10}`
`% in TeX`

`G:=Group("C4^2.138D10");`
`// GroupNames label`

`G:=SmallGroup(320,1342);`
`// by ID`

`G=gap.SmallGroup(320,1342);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,346,136,12550]);`
`// Polycyclic`

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

׿
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