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## G = C20.(C4○D4)  order 320 = 26·5

### 32nd non-split extension by C20 of C4○D4 acting via C4○D4/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20.(C4○D4)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C4⋊C4⋊7D5 — C20.(C4○D4)
 Lower central C5 — C10 — C2×C20 — C20.(C4○D4)
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C20.(C4○D4)
G = < a,b,c,d | a10=b4=1, c4=b2, d2=a5, bab-1=cac-1=dad-1=a-1, cbc-1=a5b, bd=db, dcd-1=b2c3 >

Subgroups: 382 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C10.D8, C10.Q16, C20.44D4, D101C8, C5×C4.Q8, C4⋊C47D5, D102Q8, C20.(C4○D4)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, C22×D5, C23.20D4, C4○D20, D4×D5, Q82D5, D10.13D4, SD16⋊D5, SD163D5, C20.(C4○D4)

Smallest permutation representation of C20.(C4○D4)
On 160 points
Generators in S160
```(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 35 17 45)(2 34 18 44)(3 33 19 43)(4 32 20 42)(5 31 11 41)(6 40 12 50)(7 39 13 49)(8 38 14 48)(9 37 15 47)(10 36 16 46)(21 158 140 143)(22 157 131 142)(23 156 132 141)(24 155 133 150)(25 154 134 149)(26 153 135 148)(27 152 136 147)(28 151 137 146)(29 160 138 145)(30 159 139 144)(51 62 86 77)(52 61 87 76)(53 70 88 75)(54 69 89 74)(55 68 90 73)(56 67 81 72)(57 66 82 71)(58 65 83 80)(59 64 84 79)(60 63 85 78)(91 107 126 112)(92 106 127 111)(93 105 128 120)(94 104 129 119)(95 103 130 118)(96 102 121 117)(97 101 122 116)(98 110 123 115)(99 109 124 114)(100 108 125 113)
(1 60 40 63 17 85 50 78)(2 59 31 62 18 84 41 77)(3 58 32 61 19 83 42 76)(4 57 33 70 20 82 43 75)(5 56 34 69 11 81 44 74)(6 55 35 68 12 90 45 73)(7 54 36 67 13 89 46 72)(8 53 37 66 14 88 47 71)(9 52 38 65 15 87 48 80)(10 51 39 64 16 86 49 79)(21 103 143 100 140 118 158 125)(22 102 144 99 131 117 159 124)(23 101 145 98 132 116 160 123)(24 110 146 97 133 115 151 122)(25 109 147 96 134 114 152 121)(26 108 148 95 135 113 153 130)(27 107 149 94 136 112 154 129)(28 106 150 93 137 111 155 128)(29 105 141 92 138 120 156 127)(30 104 142 91 139 119 157 126)
(1 118 6 113)(2 117 7 112)(3 116 8 111)(4 115 9 120)(5 114 10 119)(11 109 16 104)(12 108 17 103)(13 107 18 102)(14 106 19 101)(15 105 20 110)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 124 46 129)(42 123 47 128)(43 122 48 127)(44 121 49 126)(45 130 50 125)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 145 66 150)(62 144 67 149)(63 143 68 148)(64 142 69 147)(65 141 70 146)(71 155 76 160)(72 154 77 159)(73 153 78 158)(74 152 79 157)(75 151 80 156)```

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 C4○D8 C4○D20 C8.C22 Q8⋊2D5 D4×D5 SD16⋊D5 SD16⋊3D5 kernel C20.(C4○D4) C10.D8 C10.Q16 C20.44D4 D10⋊1C8 C5×C4.Q8 C4⋊C4⋊7D5 D10⋊2Q8 C2×Dic5 C22×D5 C4.Q8 C20 C4⋊C4 C2×C8 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 4 4 2 4 8 1 2 2 4 4

Matrix representation of C20.(C4○D4) in GL4(𝔽41) generated by

 1 34 0 0 7 34 0 0 0 0 1 0 0 0 0 1
,
 1 34 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
 24 38 0 0 1 17 0 0 0 0 38 0 0 0 16 27
,
 9 19 0 0 0 32 0 0 0 0 30 5 0 0 17 11
`G:=sub<GL(4,GF(41))| [1,7,0,0,34,34,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,9,0,0,0,0,9],[24,1,0,0,38,17,0,0,0,0,38,16,0,0,0,27],[9,0,0,0,19,32,0,0,0,0,30,17,0,0,5,11] >;`

C20.(C4○D4) in GAP, Magma, Sage, TeX

`C_{20}.(C_4\circ D_4)`
`% in TeX`

`G:=Group("C20.(C4oD4)");`
`// GroupNames label`

`G:=SmallGroup(320,494);`
`// by ID`

`G=gap.SmallGroup(320,494);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,100,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=a^5,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=b^2*c^3>;`
`// generators/relations`

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