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## G = C42.174D10order 320 = 26·5

### 174th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.174D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.174D10
 Lower central C5 — C2×C10 — C42.174D10
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.174D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 654 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C42 [×3], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×16], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8, C4⋊Q8 [×3], Dic10 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C23.41C23, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×10], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×C10 [×2], C20.6Q8, C42⋊D5, C20⋊Q8, Dic5.Q8 [×2], C4.Dic10, D5×C4⋊C4, C4⋊C47D5, D10⋊Q8 [×2], Dic5⋊Q8 [×2], D103Q8 [×2], C5×C4⋊Q8, C42.174D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, C22×D5 [×7], C23.41C23, Q8×D5 [×2], C23×D5, D46D10, C2×Q8×D5, Q8.10D10, C42.174D10

Smallest permutation representation of C42.174D10
On 160 points
Generators in S160
```(1 156 44 121)(2 122 45 157)(3 158 46 123)(4 124 47 159)(5 160 48 125)(6 126 49 141)(7 142 50 127)(8 128 51 143)(9 144 52 129)(10 130 53 145)(11 146 54 131)(12 132 55 147)(13 148 56 133)(14 134 57 149)(15 150 58 135)(16 136 59 151)(17 152 60 137)(18 138 41 153)(19 154 42 139)(20 140 43 155)(21 102 76 83)(22 84 77 103)(23 104 78 85)(24 86 79 105)(25 106 80 87)(26 88 61 107)(27 108 62 89)(28 90 63 109)(29 110 64 91)(30 92 65 111)(31 112 66 93)(32 94 67 113)(33 114 68 95)(34 96 69 115)(35 116 70 97)(36 98 71 117)(37 118 72 99)(38 100 73 119)(39 120 74 81)(40 82 75 101)
(1 33 54 78)(2 79 55 34)(3 35 56 80)(4 61 57 36)(5 37 58 62)(6 63 59 38)(7 39 60 64)(8 65 41 40)(9 21 42 66)(10 67 43 22)(11 23 44 68)(12 69 45 24)(13 25 46 70)(14 71 47 26)(15 27 48 72)(16 73 49 28)(17 29 50 74)(18 75 51 30)(19 31 52 76)(20 77 53 32)(81 152 110 127)(82 128 111 153)(83 154 112 129)(84 130 113 155)(85 156 114 131)(86 132 115 157)(87 158 116 133)(88 134 117 159)(89 160 118 135)(90 136 119 141)(91 142 120 137)(92 138 101 143)(93 144 102 139)(94 140 103 145)(95 146 104 121)(96 122 105 147)(97 148 106 123)(98 124 107 149)(99 150 108 125)(100 126 109 151)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 69 31 79)(22 78 32 68)(23 67 33 77)(24 76 34 66)(25 65 35 75)(26 74 36 64)(27 63 37 73)(28 72 38 62)(29 61 39 71)(30 70 40 80)(41 46 51 56)(42 55 52 45)(43 44 53 54)(47 60 57 50)(48 49 58 59)(81 117 91 107)(82 106 92 116)(83 115 93 105)(84 104 94 114)(85 113 95 103)(86 102 96 112)(87 111 97 101)(88 120 98 110)(89 109 99 119)(90 118 100 108)(121 130 131 140)(122 139 132 129)(123 128 133 138)(124 137 134 127)(125 126 135 136)(141 150 151 160)(142 159 152 149)(143 148 153 158)(144 157 154 147)(145 146 155 156)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 D10 D10 D10 2+ 1+4 2- 1+4 Q8×D5 D4⋊6D10 Q8.10D10 kernel C42.174D10 C20.6Q8 C42⋊D5 C20⋊Q8 Dic5.Q8 C4.Dic10 D5×C4⋊C4 C4⋊C4⋊7D5 D10⋊Q8 Dic5⋊Q8 D10⋊3Q8 C5×C4⋊Q8 C4×D5 C4⋊Q8 C42 C4⋊C4 C2×Q8 C10 C10 C4 C2 C2 # reps 1 1 1 1 2 1 1 1 2 2 2 1 4 2 2 8 4 1 1 4 4 4

Matrix representation of C42.174D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 26 0 0 0 0 0 0 26 0 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 24 1 0 0 0 0 40 17 0 0 0 0 0 0 24 1 0 0 0 0 40 17
,
 27 7 0 0 0 0 7 14 0 0 0 0 0 0 34 34 27 27 0 0 7 1 14 2 0 0 7 7 7 7 0 0 34 40 34 40
,
 14 34 0 0 0 0 34 27 0 0 0 0 0 0 34 34 27 27 0 0 1 7 2 14 0 0 7 7 7 7 0 0 40 34 40 34

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,11,0,0,0,0,0,0,11,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[27,7,0,0,0,0,7,14,0,0,0,0,0,0,34,7,7,34,0,0,34,1,7,40,0,0,27,14,7,34,0,0,27,2,7,40],[14,34,0,0,0,0,34,27,0,0,0,0,0,0,34,1,7,40,0,0,34,7,7,34,0,0,27,2,7,40,0,0,27,14,7,34] >;`

C42.174D10 in GAP, Magma, Sage, TeX

`C_4^2._{174}D_{10}`
`% in TeX`

`G:=Group("C4^2.174D10");`
`// GroupNames label`

`G:=SmallGroup(320,1401);`
`// by ID`

`G=gap.SmallGroup(320,1401);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^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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