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## G = C42.113D10order 320 = 26·5

### 113rd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.113D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.113D10
 Lower central C5 — C2×C10 — C42.113D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.113D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 838 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], Dic5 [×6], C20 [×2], C20 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10 [×6], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×6], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.47C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20.6Q8, C4×D20, C23.D10 [×2], D10⋊D4 [×2], D5×C4⋊C4, C4⋊C47D5, C4×C5⋊D4 [×2], C23.23D10 [×2], C202D4 [×2], D4×C20, C42.113D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.47C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D5×C4○D4, C42.113D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 D4⋊6D10 D5×C4○D4 kernel C42.113D10 C20.6Q8 C4×D20 C23.D10 D10⋊D4 D5×C4⋊C4 C4⋊C4⋊7D5 C4×C5⋊D4 C23.23D10 C20⋊2D4 D4×C20 C4×D4 C20 D10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C10 C2 C2 # reps 1 1 1 2 2 1 1 2 2 2 1 2 4 4 2 4 2 4 2 16 1 4 4

Matrix representation of C42.113D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 32
,
 39 28 0 0 13 2 0 0 0 0 9 0 0 0 0 9
,
 28 28 0 0 13 32 0 0 0 0 0 9 0 0 9 0
,
 21 20 0 0 23 20 0 0 0 0 0 32 0 0 32 0
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,9,0,0,0,0,9],[28,13,0,0,28,32,0,0,0,0,0,9,0,0,9,0],[21,23,0,0,20,20,0,0,0,0,0,32,0,0,32,0] >;`

C42.113D10 in GAP, Magma, Sage, TeX

`C_4^2._{113}D_{10}`
`% in TeX`

`G:=Group("C4^2.113D10");`
`// GroupNames label`

`G:=SmallGroup(320,1230);`
`// by ID`

`G=gap.SmallGroup(320,1230);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,570,12550]);`
`// Polycyclic`

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

׿
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