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## G = C42.125D10order 320 = 26·5

### 125th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.125D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — C42.125D10
 Lower central C5 — C10 — C42.125D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.125D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 718 in 266 conjugacy classes, 151 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C42⋊C2, C4×Q8, C4×Q8, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.32C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, C4×Dic10, C42⋊D5, Dic53Q8, C4⋊C47D5, Q8×Dic5, Q8×C20, C2×Q8×D5, C42.125D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2- 1+4, C4×D5, C22×D5, C23.32C23, C2×C4×D5, C23×D5, D5×C22×C4, Q8.10D10, D4.10D10, C42.125D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4N 4O ··· 4AB 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 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 C4 D5 D10 D10 D10 C4×D5 2- 1+4 Q8.10D10 D4.10D10 kernel C42.125D10 C4×Dic10 C42⋊D5 Dic5⋊3Q8 C4⋊C4⋊7D5 Q8×Dic5 Q8×C20 C2×Q8×D5 Q8×D5 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C10 C2 C2 # reps 1 3 3 3 3 1 1 1 16 2 6 6 2 16 2 4 4

Matrix representation of C42.125D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 11 0 1 0 0 0 0 11 0 1 0 0 1 0 30 0 0 0 0 1 0 30
,
 9 0 0 0 0 0 0 9 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
,
 1 6 0 0 0 0 35 6 0 0 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 1 34 0 0 0 0 7 34 0 0
,
 1 0 0 0 0 0 35 40 0 0 0 0 0 0 0 0 7 40 0 0 0 0 7 34 0 0 34 1 0 0 0 0 34 7 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,0,1,0,0,0,0,11,0,1,0,0,1,0,30,0,0,0,0,1,0,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,40,34,0,0,0,0,7,7,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,7,7,0,0,0,0,40,34,0,0] >;`

C42.125D10 in GAP, Magma, Sage, TeX

`C_4^2._{125}D_{10}`
`% in TeX`

`G:=Group("C4^2.125D10");`
`// GroupNames label`

`G:=SmallGroup(320,1244);`
`// by ID`

`G=gap.SmallGroup(320,1244);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,1123,80,12550]);`
`// 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,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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