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## G = C42.64D10order 320 = 26·5

### 64th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.64D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊4D4 — C42.64D10
 Lower central C5 — C10 — C2×C20 — C42.64D10
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.64D10
G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 734 in 144 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4.4D4, C41D4, C2×D8, C2×SD16, C52C8, D20, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C83D4, C2×C52C8, D4⋊D5, Q8⋊D5, C4×C20, C5×C22⋊C4, C2×D20, C2×D20, D4×C10, Q8×C10, C42.D5, C204D4, C2×D4⋊D5, C2×Q8⋊D5, C5×C4.4D4, C42.64D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C41D4, C8⋊C22, C5⋊D4, C22×D5, C83D4, D4×D5, C2×C5⋊D4, C20⋊D4, D4⋊D10, C42.64D10

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 8 40 40 2 2 4 4 8 2 2 20 20 20 20 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 C5⋊D4 C8⋊C22 D4×D5 D4⋊D10 kernel C42.64D10 C42.D5 C20⋊4D4 C2×D4⋊D5 C2×Q8⋊D5 C5×C4.4D4 C5⋊2C8 C2×C20 C4.4D4 C42 C2×D4 C2×Q8 C2×C4 C10 C4 C2 # reps 1 1 1 2 2 1 4 2 2 2 2 2 8 2 4 8

Matrix representation of C42.64D10 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 4 5 0 0 0 0 0 0 13 37 0 0 0 0 0 0 0 0 25 0 36 36 0 0 0 0 0 25 36 5 0 0 0 0 5 5 16 0 0 0 0 0 5 36 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0
,
 34 34 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 34 34 0 0 0 0 0 0 1 7 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,5,37,0,0,0,0,0,0,0,0,25,0,5,5,0,0,0,0,0,25,5,36,0,0,0,0,36,36,16,0,0,0,0,0,36,5,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,40,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;`

C42.64D10 in GAP, Magma, Sage, TeX

`C_4^2._{64}D_{10}`
`% in TeX`

`G:=Group("C4^2.64D10");`
`// GroupNames label`

`G:=SmallGroup(320,685);`
`// by ID`

`G=gap.SmallGroup(320,685);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,555,1123,297,136,12550]);`
`// Polycyclic`

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

׿
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