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## G = C42.189D10order 320 = 26·5

### 9th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.189D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.189D10
 Lower central C5 — C2×C10 — C42.189D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C42.189D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 798 in 234 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×14], C22, C22 [×10], C5, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×3], C10 [×3], C10, C42, C42 [×5], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×4], Dic5 [×4], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C10 [×3], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2, C422C2, Dic10 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×6], C22×D5 [×2], C22×C10, C23.36C23, C4×Dic5 [×5], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×6], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×4], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], D5×C42, C422D5, C23.11D10, Dic54D4 [×2], D10.12D4, D10⋊D4, Dic5.5D4, Dic53Q8, Dic5.Q8, C4⋊C47D5, D208C4, D10.13D4, D10⋊Q8, C5×C422C2, C42.189D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C23×D5, D5×C4○D4 [×3], C42.189D10

Smallest permutation representation of C42.189D10
On 160 points
Generators in S160
```(1 108 33 140)(2 114 34 81)(3 110 35 132)(4 116 36 83)(5 102 37 134)(6 118 38 85)(7 104 39 136)(8 120 40 87)(9 106 31 138)(10 112 32 89)(11 135 50 103)(12 86 41 119)(13 137 42 105)(14 88 43 111)(15 139 44 107)(16 90 45 113)(17 131 46 109)(18 82 47 115)(19 133 48 101)(20 84 49 117)(21 100 143 55)(22 64 144 79)(23 92 145 57)(24 66 146 71)(25 94 147 59)(26 68 148 73)(27 96 149 51)(28 70 150 75)(29 98 141 53)(30 62 142 77)(52 127 97 155)(54 129 99 157)(56 121 91 159)(58 123 93 151)(60 125 95 153)(61 156 76 128)(63 158 78 130)(65 160 80 122)(67 152 72 124)(69 154 74 126)
(1 95 16 73)(2 51 17 69)(3 97 18 75)(4 53 19 61)(5 99 20 77)(6 55 11 63)(7 91 12 79)(8 57 13 65)(9 93 14 71)(10 59 15 67)(21 135 158 118)(22 104 159 86)(23 137 160 120)(24 106 151 88)(25 139 152 112)(26 108 153 90)(27 131 154 114)(28 110 155 82)(29 133 156 116)(30 102 157 84)(31 58 43 66)(32 94 44 72)(33 60 45 68)(34 96 46 74)(35 52 47 70)(36 98 48 76)(37 54 49 62)(38 100 50 78)(39 56 41 64)(40 92 42 80)(81 149 109 126)(83 141 101 128)(85 143 103 130)(87 145 105 122)(89 147 107 124)(111 146 138 123)(113 148 140 125)(115 150 132 127)(117 142 134 129)(119 144 136 121)
(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)
(1 129 33 157)(2 156 34 128)(3 127 35 155)(4 154 36 126)(5 125 37 153)(6 152 38 124)(7 123 39 151)(8 160 40 122)(9 121 31 159)(10 158 32 130)(11 25 50 147)(12 146 41 24)(13 23 42 145)(14 144 43 22)(15 21 44 143)(16 142 45 30)(17 29 46 141)(18 150 47 28)(19 27 48 149)(20 148 49 26)(51 83 96 116)(52 115 97 82)(53 81 98 114)(54 113 99 90)(55 89 100 112)(56 111 91 88)(57 87 92 120)(58 119 93 86)(59 85 94 118)(60 117 95 84)(61 109 76 131)(62 140 77 108)(63 107 78 139)(64 138 79 106)(65 105 80 137)(66 136 71 104)(67 103 72 135)(68 134 73 102)(69 101 74 133)(70 132 75 110)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D5×C4○D4 kernel C42.189D10 D5×C42 C42⋊2D5 C23.11D10 Dic5⋊4D4 D10.12D4 D10⋊D4 Dic5.5D4 Dic5⋊3Q8 Dic5.Q8 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 D10⋊Q8 C5×C42⋊2C2 C42⋊2C2 Dic5 D10 C42 C22⋊C4 C4⋊C4 C2 # reps 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 8 4 2 6 6 12

Matrix representation of C42.189D10 in GL6(𝔽41)

 0 1 0 0 0 0 40 0 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 0 9
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 39 0 0 0 0 0 9
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 6 0 0 0 0 35 6 0 0 0 0 0 0 1 0 0 0 0 0 32 40
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 6 1 0 0 0 0 0 0 1 23 0 0 0 0 32 40

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,39,9],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,35,0,0,0,0,6,6,0,0,0,0,0,0,1,32,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,6,0,0,0,0,0,1,0,0,0,0,0,0,1,32,0,0,0,0,23,40] >;`

C42.189D10 in GAP, Magma, Sage, TeX

`C_4^2._{189}D_{10}`
`% in TeX`

`G:=Group("C4^2.189D10");`
`// GroupNames label`

`G:=SmallGroup(320,1378);`
`// by ID`

`G=gap.SmallGroup(320,1378);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,794,297,12550]);`
`// Polycyclic`

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

׿
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