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## G = C42.82D10order 320 = 26·5

### 82nd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.82D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — C42.82D10
 Lower central C5 — C10 — C2×C20 — C42.82D10
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C42.82D10
G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >

Subgroups: 414 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C52C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C42.28C22, C2×C52C8, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, Q8×C10, C42.D5, D206C4, C10.Q16, C4.D20, C5×C4⋊Q8, C42.82D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C42.28C22, Q82D5, C2×C5⋊D4, D4.D10, C20.C23, C20.23D4, C42.82D10

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 40 2 2 4 4 8 8 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C5⋊D4 C8⋊C22 C8.C22 Q8⋊2D5 D4.D10 C20.C23 kernel C42.82D10 C42.D5 D20⋊6C4 C10.Q16 C4.D20 C5×C4⋊Q8 C2×C20 C4⋊Q8 C20 C42 C4⋊C4 C2×C4 C10 C10 C4 C2 C2 # reps 1 1 2 2 1 1 2 2 4 2 4 8 1 1 4 4 4

Matrix representation of C42.82D10 in GL8(𝔽41)

 13 33 35 35 0 0 0 0 8 28 6 0 0 0 0 0 0 37 21 8 0 0 0 0 4 4 40 20 0 0 0 0 0 0 0 0 0 0 18 5 0 0 0 0 0 0 1 23 0 0 0 0 23 36 0 0 0 0 0 0 40 18 0 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0
,
 38 3 0 21 0 0 0 0 38 17 23 20 0 0 0 0 9 13 21 38 0 0 0 0 0 37 26 6 0 0 0 0 0 0 0 0 4 27 37 7 0 0 0 0 30 31 26 3 0 0 0 0 37 7 37 14 0 0 0 0 26 3 11 10
,
 3 38 0 20 0 0 0 0 17 38 23 3 0 0 0 0 13 9 21 24 0 0 0 0 37 0 26 20 0 0 0 0 0 0 0 0 10 27 3 34 0 0 0 0 21 31 29 38 0 0 0 0 38 7 10 27 0 0 0 0 12 3 21 31

`G:=sub<GL(8,GF(41))| [13,8,0,4,0,0,0,0,33,28,37,4,0,0,0,0,35,6,21,40,0,0,0,0,35,0,8,20,0,0,0,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18,0,0,0,0,18,1,0,0,0,0,0,0,5,23,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[38,38,9,0,0,0,0,0,3,17,13,37,0,0,0,0,0,23,21,26,0,0,0,0,21,20,38,6,0,0,0,0,0,0,0,0,4,30,37,26,0,0,0,0,27,31,7,3,0,0,0,0,37,26,37,11,0,0,0,0,7,3,14,10],[3,17,13,37,0,0,0,0,38,38,9,0,0,0,0,0,0,23,21,26,0,0,0,0,20,3,24,20,0,0,0,0,0,0,0,0,10,21,38,12,0,0,0,0,27,31,7,3,0,0,0,0,3,29,10,21,0,0,0,0,34,38,27,31] >;`

C42.82D10 in GAP, Magma, Sage, TeX

`C_4^2._{82}D_{10}`
`% in TeX`

`G:=Group("C4^2.82D10");`
`// GroupNames label`

`G:=SmallGroup(320,716);`
`// by ID`

`G=gap.SmallGroup(320,716);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,100,1123,297,136,12550]);`
`// Polycyclic`

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

׿
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