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## G = C42.274D10order 320 = 26·5

### 33rd non-split extension by C42 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.274D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C4×Dic10 — C42.274D10
 Lower central C5 — C2×C10 — C42.274D10
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.274D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >

Subgroups: 606 in 222 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×8], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], Dic10 [×8], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×C10, C23.37C23, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×8], C23.D5 [×4], C4×C20 [×2], C4×C20 [×2], C2×Dic10 [×4], C22×C20, C22×C20 [×2], C4×Dic10 [×4], C202Q8 [×2], C20.6Q8 [×2], C20.48D4 [×4], C23.21D10 [×2], C2×C4×C20, C42.274D10
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], Dic10 [×4], C22×D5 [×7], C23.37C23, C2×Dic10 [×6], C4○D20 [×4], C23×D5, C22×Dic10, C2×C4○D20 [×2], C42.274D10

Smallest permutation representation of C42.274D10
On 160 points
Generators in S160
```(1 48 8 43)(2 49 9 44)(3 50 10 45)(4 46 6 41)(5 47 7 42)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 79 39 74)(32 80 40 75)(33 76 36 71)(34 77 37 72)(35 78 38 73)(81 132 91 145)(82 133 92 146)(83 134 93 147)(84 135 94 148)(85 136 95 149)(86 137 96 150)(87 138 97 141)(88 139 98 142)(89 140 99 143)(90 131 100 144)(101 113 153 126)(102 114 154 127)(103 115 155 128)(104 116 156 129)(105 117 157 130)(106 118 158 121)(107 119 159 122)(108 120 160 123)(109 111 151 124)(110 112 152 125)
(1 35 13 23)(2 31 14 24)(3 32 15 25)(4 33 11 21)(5 34 12 22)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)(81 117 86 112)(82 118 87 113)(83 119 88 114)(84 120 89 115)(85 111 90 116)(91 130 96 125)(92 121 97 126)(93 122 98 127)(94 123 99 128)(95 124 100 129)(101 146 106 141)(102 147 107 142)(103 148 108 143)(104 149 109 144)(105 150 110 145)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(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)
(1 100 18 85)(2 94 19 89)(3 98 20 83)(4 92 16 87)(5 96 17 81)(6 82 11 97)(7 86 12 91)(8 90 13 95)(9 84 14 99)(10 88 15 93)(21 126 36 118)(22 130 37 112)(23 124 38 116)(24 128 39 120)(25 122 40 114)(26 113 33 121)(27 117 34 125)(28 111 35 129)(29 115 31 123)(30 119 32 127)(41 146 56 138)(42 150 57 132)(43 144 58 136)(44 148 59 140)(45 142 60 134)(46 133 51 141)(47 137 52 145)(48 131 53 149)(49 135 54 143)(50 139 55 147)(61 101 76 158)(62 105 77 152)(63 109 78 156)(64 103 79 160)(65 107 80 154)(66 153 71 106)(67 157 72 110)(68 151 73 104)(69 155 74 108)(70 159 75 102)```

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 Dic10 C4○D20 kernel C42.274D10 C4×Dic10 C20⋊2Q8 C20.6Q8 C20.48D4 C23.21D10 C2×C4×C20 C2×C20 C2×C42 C20 C42 C22×C4 C2×C4 C4 # reps 1 4 2 2 4 2 1 4 2 8 8 6 16 32

Matrix representation of C42.274D10 in GL4(𝔽41) generated by

 32 0 0 0 18 9 0 0 0 0 2 28 0 0 13 39
,
 9 0 0 0 0 9 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 39 40 0 0 0 0 40 35 0 0 6 35
,
 9 9 0 0 23 32 0 0 0 0 4 14 0 0 31 37
`G:=sub<GL(4,GF(41))| [32,18,0,0,0,9,0,0,0,0,2,13,0,0,28,39],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[1,39,0,0,0,40,0,0,0,0,40,6,0,0,35,35],[9,23,0,0,9,32,0,0,0,0,4,31,0,0,14,37] >;`

C42.274D10 in GAP, Magma, Sage, TeX

`C_4^2._{274}D_{10}`
`% in TeX`

`G:=Group("C4^2.274D10");`
`// GroupNames label`

`G:=SmallGroup(320,1142);`
`// by ID`

`G=gap.SmallGroup(320,1142);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,675,12550]);`
`// Polycyclic`

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

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