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## G = C10.(C4○D8)  order 320 = 26·5

### 17th non-split extension by C10 of C4○D8 acting via C4○D8/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C10.(C4○D8)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C23.21D10 — C10.(C4○D8)
 Lower central C5 — C10 — C2×C20 — C10.(C4○D8)
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for C10.(C4○D8)
G = < a,b,c,d | a10=b8=d2=1, c2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=b-1, dbd=a5b, dcd=b4c >

Subgroups: 286 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C23.20D4, C2×C52C8, C4×Dic5, C4⋊Dic5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C10.D8, C20.Q8, C20.55D4, Q8⋊Dic5, C23.21D10, C5×C22⋊Q8, C10.(C4○D8)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, C5⋊D4, C22×D5, C23.20D4, D42D5, C2×C5⋊D4, C23.18D10, C20.C23, D4.8D10, C10.(C4○D8)

Smallest permutation representation of C10.(C4○D8)
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)
(1 138 32 158 25 150 45 17)(2 137 33 157 26 149 46 16)(3 136 34 156 27 148 47 15)(4 135 35 155 28 147 48 14)(5 134 36 154 29 146 49 13)(6 133 37 153 30 145 50 12)(7 132 38 152 21 144 41 11)(8 131 39 151 22 143 42 20)(9 140 40 160 23 142 43 19)(10 139 31 159 24 141 44 18)(51 94 83 126 64 106 71 114)(52 93 84 125 65 105 72 113)(53 92 85 124 66 104 73 112)(54 91 86 123 67 103 74 111)(55 100 87 122 68 102 75 120)(56 99 88 121 69 101 76 119)(57 98 89 130 70 110 77 118)(58 97 90 129 61 109 78 117)(59 96 81 128 62 108 79 116)(60 95 82 127 63 107 80 115)
(1 93 6 98)(2 92 7 97)(3 91 8 96)(4 100 9 95)(5 99 10 94)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
(11 16)(12 17)(13 18)(14 19)(15 20)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 101)(100 102)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 121)(120 122)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)```

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

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

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

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 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 2 2 2 2 8 8 20 20 20 20 2 2 20 20 20 20 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

47 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 C5⋊D4 C8.C22 D4⋊2D5 C20.C23 D4.8D10 kernel C10.(C4○D8) C10.D8 C20.Q8 C20.55D4 Q8⋊Dic5 C23.21D10 C5×C22⋊Q8 C2×C20 C22×C10 C22⋊Q8 C20 C4⋊C4 C22×C4 C2×Q8 C10 C2×C4 C23 C10 C4 C2 C2 # reps 1 1 1 1 2 1 1 1 1 2 4 2 2 2 4 4 4 1 4 4 4

Matrix representation of C10.(C4○D8) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 37 0 0 0 0 0 23 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 22 0 0 0 0 9 34 0 0 0 0 0 0 30 4 0 0 0 0 11 11 0 0 0 0 0 0 38 0 0 0 0 0 21 27
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 30 4 0 0 0 0 11 11 0 0 0 0 0 0 40 22 0 0 0 0 0 1
,
 1 3 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 15 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,23,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,9,0,0,0,0,22,34,0,0,0,0,0,0,30,11,0,0,0,0,4,11,0,0,0,0,0,0,38,21,0,0,0,0,0,27],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,11,0,0,0,0,4,11,0,0,0,0,0,0,40,0,0,0,0,0,22,1],[1,0,0,0,0,0,3,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,0,40] >;`

C10.(C4○D8) in GAP, Magma, Sage, TeX

`C_{10}.(C_4\circ D_8)`
`% in TeX`

`G:=Group("C10.(C4oD8)");`
`// GroupNames label`

`G:=SmallGroup(320,672);`
`// by ID`

`G=gap.SmallGroup(320,672);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,219,184,1123,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^8=d^2=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a^5*b,d*c*d=b^4*c>;`
`// generators/relations`

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