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## G = C42.171D10order 320 = 26·5

### 171st non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.171D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — C42.171D10
 Lower central C5 — C2×C10 — C42.171D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.171D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 990 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4, C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×10], C23 [×3], D5 [×4], C10, C10 [×2], C42, C42, C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×5], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×8], C2×C10, C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10 [×6], C4×D5 [×4], C4×D5 [×8], D20 [×6], C2×Dic5, C2×Dic5 [×4], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22×D5 [×2], C23.38C23, C4×Dic5, C10.D4 [×6], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], Q8×D5 [×4], Q82D5 [×4], Q8×C10 [×2], C42⋊D5, C4.D20, D10.13D4 [×4], D10⋊Q8 [×4], Dic5⋊Q8, C20.23D4, C5×C4⋊Q8, C2×Q8×D5, C2×Q82D5, C42.171D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- 1+4 [×2], C22×D5 [×7], C23.38C23, D4×D5 [×2], C23×D5, C2×D4×D5, Q8.10D10 [×2], C42.171D10

Smallest permutation representation of C42.171D10
On 160 points
Generators in S160
```(1 44 11 54)(2 55 12 45)(3 46 13 56)(4 57 14 47)(5 48 15 58)(6 59 16 49)(7 50 17 60)(8 41 18 51)(9 52 19 42)(10 43 20 53)(21 125 31 135)(22 136 32 126)(23 127 33 137)(24 138 34 128)(25 129 35 139)(26 140 36 130)(27 131 37 121)(28 122 38 132)(29 133 39 123)(30 124 40 134)(61 101 71 111)(62 112 72 102)(63 103 73 113)(64 114 74 104)(65 105 75 115)(66 116 76 106)(67 107 77 117)(68 118 78 108)(69 109 79 119)(70 120 80 110)(81 151 91 141)(82 142 92 152)(83 153 93 143)(84 144 94 154)(85 155 95 145)(86 146 96 156)(87 157 97 147)(88 148 98 158)(89 159 99 149)(90 150 100 160)
(1 97 61 134)(2 135 62 98)(3 99 63 136)(4 137 64 100)(5 81 65 138)(6 139 66 82)(7 83 67 140)(8 121 68 84)(9 85 69 122)(10 123 70 86)(11 87 71 124)(12 125 72 88)(13 89 73 126)(14 127 74 90)(15 91 75 128)(16 129 76 92)(17 93 77 130)(18 131 78 94)(19 95 79 132)(20 133 80 96)(21 112 158 55)(22 56 159 113)(23 114 160 57)(24 58 141 115)(25 116 142 59)(26 60 143 117)(27 118 144 41)(28 42 145 119)(29 120 146 43)(30 44 147 101)(31 102 148 45)(32 46 149 103)(33 104 150 47)(34 48 151 105)(35 106 152 49)(36 50 153 107)(37 108 154 51)(38 52 155 109)(39 110 156 53)(40 54 157 111)
(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)
(1 138 11 128)(2 127 12 137)(3 136 13 126)(4 125 14 135)(5 134 15 124)(6 123 16 133)(7 132 17 122)(8 121 18 131)(9 130 19 140)(10 139 20 129)(21 47 31 57)(22 56 32 46)(23 45 33 55)(24 54 34 44)(25 43 35 53)(26 52 36 42)(27 41 37 51)(28 50 38 60)(29 59 39 49)(30 48 40 58)(61 81 71 91)(62 90 72 100)(63 99 73 89)(64 88 74 98)(65 97 75 87)(66 86 76 96)(67 95 77 85)(68 84 78 94)(69 93 79 83)(70 82 80 92)(101 141 111 151)(102 150 112 160)(103 159 113 149)(104 148 114 158)(105 157 115 147)(106 146 116 156)(107 155 117 145)(108 144 118 154)(109 153 119 143)(110 142 120 152)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C ··· 4H 4I 4J 4K 4L 4M 4N 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 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 20 20 2 2 4 ··· 4 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 2- 1+4 D4×D5 Q8.10D10 kernel C42.171D10 C42⋊D5 C4.D20 D10.13D4 D10⋊Q8 Dic5⋊Q8 C20.23D4 C5×C4⋊Q8 C2×Q8×D5 C2×Q8⋊2D5 C4×D5 C4⋊Q8 C42 C4⋊C4 C2×Q8 C10 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 4 2 2 8 4 2 4 8

Matrix representation of C42.171D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 2 9 0 0 0 0 4 39 0 0 0 0 6 1 30 32 0 0 40 0 9 11
,
 32 9 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 3 18 9 0 0 0 3 18 0 9
,
 9 32 0 0 0 0 18 32 0 0 0 0 0 0 10 19 27 33 0 0 32 28 35 34 0 0 34 0 22 22 0 0 34 34 22 22
,
 32 9 0 0 0 0 23 9 0 0 0 0 0 0 28 22 0 0 0 0 37 13 0 0 0 0 34 0 22 22 0 0 1 7 32 19

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,4,6,40,0,0,9,39,1,0,0,0,0,0,30,9,0,0,0,0,32,11],[32,0,0,0,0,0,9,9,0,0,0,0,0,0,32,0,3,3,0,0,0,32,18,18,0,0,0,0,9,0,0,0,0,0,0,9],[9,18,0,0,0,0,32,32,0,0,0,0,0,0,10,32,34,34,0,0,19,28,0,34,0,0,27,35,22,22,0,0,33,34,22,22],[32,23,0,0,0,0,9,9,0,0,0,0,0,0,28,37,34,1,0,0,22,13,0,7,0,0,0,0,22,32,0,0,0,0,22,19] >;`

C42.171D10 in GAP, Magma, Sage, TeX

`C_4^2._{171}D_{10}`
`% in TeX`

`G:=Group("C4^2.171D10");`
`// GroupNames label`

`G:=SmallGroup(320,1396);`
`// by ID`

`G=gap.SmallGroup(320,1396);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,136,12550]);`
`// Polycyclic`

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

׿
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