Copied to
clipboard

## G = C42.168D10order 320 = 26·5

### 168th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.168D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C42⋊D5 — C42.168D10
 Lower central C5 — C2×C10 — C42.168D10
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for C42.168D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c-1 >

Subgroups: 846 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C41D4, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.34C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×C10, C20.6Q8, C42⋊D5, D4×Dic5, C23.18D10, C202D4, Dic5⋊D4, C5×C41D4, C42.168D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.34C24, D42D5, C23×D5, C2×D42D5, D46D10, C42.168D10

Smallest permutation representation of C42.168D10
On 160 points
Generators in S160
```(1 108 33 113)(2 114 34 109)(3 110 35 115)(4 116 36 101)(5 102 37 117)(6 118 38 103)(7 104 39 119)(8 120 40 105)(9 106 31 111)(10 112 32 107)(11 132 57 97)(12 98 58 133)(13 134 59 99)(14 100 60 135)(15 136 51 91)(16 92 52 137)(17 138 53 93)(18 94 54 139)(19 140 55 95)(20 96 56 131)(21 90 143 69)(22 70 144 81)(23 82 145 61)(24 62 146 83)(25 84 147 63)(26 64 148 85)(27 86 149 65)(28 66 150 87)(29 88 141 67)(30 68 142 89)(41 130 73 153)(42 154 74 121)(43 122 75 155)(44 156 76 123)(45 124 77 157)(46 158 78 125)(47 126 79 159)(48 160 80 127)(49 128 71 151)(50 152 72 129)
(1 41 14 64)(2 65 15 42)(3 43 16 66)(4 67 17 44)(5 45 18 68)(6 69 19 46)(7 47 20 70)(8 61 11 48)(9 49 12 62)(10 63 13 50)(21 140 158 118)(22 119 159 131)(23 132 160 120)(24 111 151 133)(25 134 152 112)(26 113 153 135)(27 136 154 114)(28 115 155 137)(29 138 156 116)(30 117 157 139)(31 71 58 83)(32 84 59 72)(33 73 60 85)(34 86 51 74)(35 75 52 87)(36 88 53 76)(37 77 54 89)(38 90 55 78)(39 79 56 81)(40 82 57 80)(91 121 109 149)(92 150 110 122)(93 123 101 141)(94 142 102 124)(95 125 103 143)(96 144 104 126)(97 127 105 145)(98 146 106 128)(99 129 107 147)(100 148 108 130)
(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 14 13)(2 12 15 9)(3 8 16 11)(4 20 17 7)(5 6 18 19)(21 124 158 142)(22 141 159 123)(23 122 160 150)(24 149 151 121)(25 130 152 148)(26 147 153 129)(27 128 154 146)(28 145 155 127)(29 126 156 144)(30 143 157 125)(31 34 58 51)(32 60 59 33)(35 40 52 57)(36 56 53 39)(37 38 54 55)(41 72 64 84)(42 83 65 71)(43 80 66 82)(44 81 67 79)(45 78 68 90)(46 89 69 77)(47 76 70 88)(48 87 61 75)(49 74 62 86)(50 85 63 73)(91 106 109 98)(92 97 110 105)(93 104 101 96)(94 95 102 103)(99 108 107 100)(111 114 133 136)(112 135 134 113)(115 120 137 132)(116 131 138 119)(117 118 139 140)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 kernel C42.168D10 C20.6Q8 C42⋊D5 D4×Dic5 C23.18D10 C20⋊2D4 Dic5⋊D4 C5×C4⋊1D4 C4⋊1D4 C20 C42 C2×D4 C10 C4 C2 # reps 1 1 1 2 4 2 4 1 2 4 2 12 2 4 8

Matrix representation of C42.168D10 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 11 0 9 0 0 0 0 0 0 11 0 9 0 0 0 0 32 0 30 0 0 0 0 0 0 32 0 30
,
 27 0 36 22 0 0 0 0 0 27 19 5 0 0 0 0 5 19 14 0 0 0 0 0 22 36 0 14 0 0 0 0 0 0 0 0 18 6 0 0 0 0 0 0 35 23 0 0 0 0 0 0 0 0 18 6 0 0 0 0 0 0 35 23
,
 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 0 0 0 40 35 0 0 0 0 0 0 6 35 0 0 0 0 40 35 0 0 0 0 0 0 6 35 0 0
,
 0 0 34 34 0 0 0 0 0 0 1 7 0 0 0 0 7 7 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 0 0 35 40 0 0 0 0 0 0 35 6 0 0 0 0 35 40 0 0 0 0 0 0 35 6 0 0

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,11,0,32,0,0,0,0,0,0,11,0,32,0,0,0,0,9,0,30,0,0,0,0,0,0,9,0,30],[27,0,5,22,0,0,0,0,0,27,19,36,0,0,0,0,36,19,14,0,0,0,0,0,22,5,0,14,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23],[0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0] >;`

C42.168D10 in GAP, Magma, Sage, TeX

`C_4^2._{168}D_{10}`
`% in TeX`

`G:=Group("C4^2.168D10");`
`// GroupNames label`

`G:=SmallGroup(320,1391);`
`// by ID`

`G=gap.SmallGroup(320,1391);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,185,12550]);`
`// Polycyclic`

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

׿
×
𝔽