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## G = C4○D20⋊9C4order 320 = 26·5

### 3rd semidirect product of C4○D20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4○D20⋊9C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C2×C4○D20 — C4○D20⋊9C4
 Lower central C5 — C10 — C20 — C4○D20⋊9C4
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for C4○D209C4
G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, dad-1=a-1, cbc=a2b9, dbd-1=a2b, dcd-1=a2b5c >

Subgroups: 606 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10 [×3], C10 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C52C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×D5, C22×C10, C23.36D4, C2×C52C8 [×2], C4.Dic5 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D206C4 [×2], C10.Q16 [×2], C2×C4.Dic5, C10×C4⋊C4, C2×C4○D20, C4○D209C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.36D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.D10, C20.C23, C4○D209C4

Smallest permutation representation of C4○D209C4
On 160 points
Generators in S160
```(1 100 11 90)(2 81 12 91)(3 82 13 92)(4 83 14 93)(5 84 15 94)(6 85 16 95)(7 86 17 96)(8 87 18 97)(9 88 19 98)(10 89 20 99)(21 103 31 113)(22 104 32 114)(23 105 33 115)(24 106 34 116)(25 107 35 117)(26 108 36 118)(27 109 37 119)(28 110 38 120)(29 111 39 101)(30 112 40 102)(41 135 51 125)(42 136 52 126)(43 137 53 127)(44 138 54 128)(45 139 55 129)(46 140 56 130)(47 121 57 131)(48 122 58 132)(49 123 59 133)(50 124 60 134)(61 150 71 160)(62 151 72 141)(63 152 73 142)(64 153 74 143)(65 154 75 144)(66 155 76 145)(67 156 77 146)(68 157 78 147)(69 158 79 148)(70 159 80 149)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)
(1 73 33 49)(2 64 34 60)(3 75 35 51)(4 66 36 42)(5 77 37 53)(6 68 38 44)(7 79 39 55)(8 70 40 46)(9 61 21 57)(10 72 22 48)(11 63 23 59)(12 74 24 50)(13 65 25 41)(14 76 26 52)(15 67 27 43)(16 78 28 54)(17 69 29 45)(18 80 30 56)(19 71 31 47)(20 62 32 58)(81 143 116 124)(82 154 117 135)(83 145 118 126)(84 156 119 137)(85 147 120 128)(86 158 101 139)(87 149 102 130)(88 160 103 121)(89 151 104 132)(90 142 105 123)(91 153 106 134)(92 144 107 125)(93 155 108 136)(94 146 109 127)(95 157 110 138)(96 148 111 129)(97 159 112 140)(98 150 113 131)(99 141 114 122)(100 152 115 133)```

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 C5⋊D4 C8⋊C22 C8.C22 D4.D10 C20.C23 kernel C4○D20⋊9C4 D20⋊6C4 C10.Q16 C2×C4.Dic5 C10×C4⋊C4 C2×C4○D20 C4○D20 C2×C20 C22×C10 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C10 C10 C2 C2 # reps 1 2 2 1 1 1 8 3 1 2 4 2 8 8 4 4 1 1 4 4

Matrix representation of C4○D209C4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 23 6 5 12 0 0 35 18 29 36 0 0 18 35 18 35 0 0 6 23 6 23
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 39 0 0 1 35 2 29 0 0 0 1 0 1 0 0 40 6 40 6
,
 40 0 0 0 0 0 9 1 0 0 0 0 0 0 0 40 0 39 0 0 40 0 39 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 32 39 0 0 0 0 0 9 0 0 0 0 0 0 2 13 34 13 0 0 28 39 28 30 0 0 15 14 39 28 0 0 27 17 13 2

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,23,35,18,6,0,0,6,18,35,23,0,0,5,29,18,6,0,0,12,36,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,40,0,0,40,35,1,6,0,0,0,2,0,40,0,0,39,29,1,6],[40,9,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,39,0,1,0,0,39,0,1,0],[32,0,0,0,0,0,39,9,0,0,0,0,0,0,2,28,15,27,0,0,13,39,14,17,0,0,34,28,39,13,0,0,13,30,28,2] >;`

C4○D209C4 in GAP, Magma, Sage, TeX

`C_4\circ D_{20}\rtimes_9C_4`
`% in TeX`

`G:=Group("C4oD20:9C4");`
`// GroupNames label`

`G:=SmallGroup(320,593);`
`// by ID`

`G=gap.SmallGroup(320,593);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,387,58,1684,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b^9,d*b*d^-1=a^2*b,d*c*d^-1=a^2*b^5*c>;`
`// generators/relations`

׿
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