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## G = C42.236D10order 320 = 26·5

### 56th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.236D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.236D10
 Lower central C5 — C2×C10 — C42.236D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.236D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=a2c9 >

Subgroups: 686 in 222 conjugacy classes, 107 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×16], C22, C22 [×4], C5, C2×C4, C2×C4 [×6], C2×C4 [×15], Q8 [×8], C23, D5 [×2], C10, C10 [×2], C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C2×Q8 [×4], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2, C42.C2, C4⋊Q8 [×2], Dic10 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C22×D5, C23.37C23, C4×Dic5, C4×Dic5 [×6], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×4], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], C20.6Q8, D5×C42, Dic53Q8 [×4], C20⋊Q8 [×2], C4⋊C47D5 [×2], D10⋊Q8 [×4], C5×C42.C2, C42.236D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], C22×D5 [×7], C23.37C23, Q8×D5 [×2], C23×D5, C2×Q8×D5, D5×C4○D4 [×2], C42.236D10

Smallest permutation representation of C42.236D10
On 160 points
Generators in S160
```(1 47 118 123)(2 124 119 48)(3 49 120 125)(4 126 101 50)(5 51 102 127)(6 128 103 52)(7 53 104 129)(8 130 105 54)(9 55 106 131)(10 132 107 56)(11 57 108 133)(12 134 109 58)(13 59 110 135)(14 136 111 60)(15 41 112 137)(16 138 113 42)(17 43 114 139)(18 140 115 44)(19 45 116 121)(20 122 117 46)(21 89 67 152)(22 153 68 90)(23 91 69 154)(24 155 70 92)(25 93 71 156)(26 157 72 94)(27 95 73 158)(28 159 74 96)(29 97 75 160)(30 141 76 98)(31 99 77 142)(32 143 78 100)(33 81 79 144)(34 145 80 82)(35 83 61 146)(36 147 62 84)(37 85 63 148)(38 149 64 86)(39 87 65 150)(40 151 66 88)
(1 79 11 69)(2 24 12 34)(3 61 13 71)(4 26 14 36)(5 63 15 73)(6 28 16 38)(7 65 17 75)(8 30 18 40)(9 67 19 77)(10 32 20 22)(21 116 31 106)(23 118 33 108)(25 120 35 110)(27 102 37 112)(29 104 39 114)(41 158 51 148)(42 86 52 96)(43 160 53 150)(44 88 54 98)(45 142 55 152)(46 90 56 100)(47 144 57 154)(48 92 58 82)(49 146 59 156)(50 94 60 84)(62 101 72 111)(64 103 74 113)(66 105 76 115)(68 107 78 117)(70 109 80 119)(81 133 91 123)(83 135 93 125)(85 137 95 127)(87 139 97 129)(89 121 99 131)(122 153 132 143)(124 155 134 145)(126 157 136 147)(128 159 138 149)(130 141 140 151)
(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 10 108 117)(2 116 109 9)(3 8 110 115)(4 114 111 7)(5 6 112 113)(11 20 118 107)(12 106 119 19)(13 18 120 105)(14 104 101 17)(15 16 102 103)(21 34 77 70)(22 69 78 33)(23 32 79 68)(24 67 80 31)(25 30 61 66)(26 65 62 29)(27 28 63 64)(35 40 71 76)(36 75 72 39)(37 38 73 74)(41 42 127 128)(43 60 129 126)(44 125 130 59)(45 58 131 124)(46 123 132 57)(47 56 133 122)(48 121 134 55)(49 54 135 140)(50 139 136 53)(51 52 137 138)(81 90 154 143)(82 142 155 89)(83 88 156 141)(84 160 157 87)(85 86 158 159)(91 100 144 153)(92 152 145 99)(93 98 146 151)(94 150 147 97)(95 96 148 149)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 4S 4T 4U 4V 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 2 ··· 2 4 4 4 4 5 5 5 5 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 Q8×D5 D5×C4○D4 kernel C42.236D10 C20.6Q8 D5×C42 Dic5⋊3Q8 C20⋊Q8 C4⋊C4⋊7D5 D10⋊Q8 C5×C42.C2 C4×D5 C42.C2 Dic5 C42 C4⋊C4 C4 C2 # reps 1 1 1 4 2 2 4 1 4 2 8 2 12 4 8

Matrix representation of C42.236D10 in GL6(𝔽41)

 9 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 1 32
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 6 0 0 0 0 34 7 0 0 0 0 0 0 40 18 0 0 0 0 9 1
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 34 6 0 0 0 0 33 7 0 0 0 0 0 0 1 23 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,9,0,0,0,0,18,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,34,33,0,0,0,0,6,7,0,0,0,0,0,0,1,0,0,0,0,0,23,40] >;`

C42.236D10 in GAP, Magma, Sage, TeX

`C_4^2._{236}D_{10}`
`% in TeX`

`G:=Group("C4^2.236D10");`
`// GroupNames label`

`G:=SmallGroup(320,1360);`
`// by ID`

`G=gap.SmallGroup(320,1360);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,570,409,80,12550]);`
`// Polycyclic`

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

׿
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