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## G = C42.106D10order 320 = 26·5

### 106th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.106D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊3Q8 — C42.106D10
 Lower central C5 — C2×C10 — C42.106D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.106D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c9 >

Subgroups: 598 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], C10 [×3], C10 [×2], C42, C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×11], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×8], C20 [×4], C20 [×3], C2×C10, C2×C10 [×6], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, Dic10 [×6], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C22.50C24, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×4], C23.D5 [×8], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C202Q8, C23.D10 [×4], Dic53Q8 [×2], C20.48D4 [×2], C23.21D10 [×2], C20.17D4 [×2], D4×C20, C42.106D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.50C24, C4○D20 [×2], D42D5 [×2], C23×D5, C2×C4○D20, C2×D42D5, D4.10D10, C42.106D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 D10 C4○D20 2- 1+4 D4⋊2D5 D4.10D10 kernel C42.106D10 C4×Dic10 C20⋊2Q8 C23.D10 Dic5⋊3Q8 C20.48D4 C23.21D10 C20.17D4 D4×C20 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C10 C4 C2 # reps 1 1 1 4 2 2 2 2 1 2 8 2 4 2 4 2 16 1 4 4

Matrix representation of C42.106D10 in GL4(𝔽41) generated by

 0 40 0 0 1 0 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 31 32
,
 0 1 0 0 1 0 0 0 0 0 21 0 0 0 10 39
,
 0 9 0 0 9 0 0 0 0 0 39 21 0 0 35 2
`G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,9,31,0,0,0,32],[0,1,0,0,1,0,0,0,0,0,21,10,0,0,0,39],[0,9,0,0,9,0,0,0,0,0,39,35,0,0,21,2] >;`

C42.106D10 in GAP, Magma, Sage, TeX

`C_4^2._{106}D_{10}`
`% in TeX`

`G:=Group("C4^2.106D10");`
`// GroupNames label`

`G:=SmallGroup(320,1214);`
`// by ID`

`G=gap.SmallGroup(320,1214);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,192,12550]);`
`// Polycyclic`

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

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