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## G = C10.(C4⋊D4)  order 320 = 26·5

### 7th non-split extension by C10 of C4⋊D4 acting via C4⋊D4/C22⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — C10.(C4⋊D4)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — C10.(C4⋊D4)
 Lower central C5 — C22×C10 — C10.(C4⋊D4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C10.(C4⋊D4)
G = < a,b,c,d | a10=b4=c4=1, d2=a5, bab-1=cac-1=a-1, ad=da, cbc-1=a5b-1, dbd-1=b-1, dcd-1=a5c-1 >

Subgroups: 742 in 170 conjugacy classes, 55 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C23.11D4, C10.D4, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4, C2×D10⋊C4, C10.(C4⋊D4)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22.D4, C4.4D4, C422C2, C22×D5, C23.11D4, C4○D20, D4×D5, D42D5, Q82D5, C422D5, D10.12D4, D10⋊D4, Dic5.5D4, D10.13D4, C4⋊C4⋊D5, C10.(C4⋊D4)

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

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4F 4G ··· 4L 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 4 ··· 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel C10.(C4⋊D4) C10.10C42 C5×C2.C42 C2×C10.D4 C2×D10⋊C4 C2×Dic5 C22×D5 C2.C42 C2×C10 C22×C4 C22 C22 C22 C22 # reps 1 2 1 1 3 2 2 2 10 6 24 4 2 2

Matrix representation of C10.(C4⋊D4) in GL6(𝔽41)

 6 6 0 0 0 0 35 1 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 13 0 0 0 0 25 39 0 0 0 0 0 0 22 19 0 0 0 0 9 19 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 0 0 0 0 0 13 32 0 0 0 0 0 0 3 3 0 0 0 0 24 38 0 0 0 0 0 0 15 37 0 0 0 0 36 26
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 24 40 0 0 0 0 1 17 0 0 0 0 0 0 15 37 0 0 0 0 15 26

`G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,22,9,0,0,0,0,19,19,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,13,0,0,0,0,0,32,0,0,0,0,0,0,3,24,0,0,0,0,3,38,0,0,0,0,0,0,15,36,0,0,0,0,37,26],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,15,15,0,0,0,0,37,26] >;`

C10.(C4⋊D4) in GAP, Magma, Sage, TeX

`C_{10}.(C_4\rtimes D_4)`
`% in TeX`

`G:=Group("C10.(C4:D4)");`
`// GroupNames label`

`G:=SmallGroup(320,302);`
`// by ID`

`G=gap.SmallGroup(320,302);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,590,387,100,12550]);`
`// Polycyclic`

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

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