Copied to
clipboard

## G = C42.229D10order 320 = 26·5

### 49th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.229D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.229D10
 Lower central C5 — C2×C10 — C42.229D10
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C42.229D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 718 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×2], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, Dic5 [×7], C20 [×4], C20 [×3], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C4×D5 [×6], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5, C22×C10 [×2], C23.36C23, C4×Dic5 [×3], C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×3], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, D5×C42, C23.D10 [×2], D10.12D4 [×2], C4.Dic10, D102Q8, C23.21D10 [×2], C4×C5⋊D4 [×2], C20.17D4, C202D4, D4×C20, C42.229D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×2], D42D5 [×2], C23×D5, C2×C4○D20, C2×D42D5, D5×C4○D4, C42.229D10

Smallest permutation representation of C42.229D10
On 160 points
Generators in S160
```(1 108 33 135)(2 114 34 81)(3 110 35 137)(4 116 36 83)(5 102 37 139)(6 118 38 85)(7 104 39 131)(8 120 40 87)(9 106 31 133)(10 112 32 89)(11 134 49 107)(12 90 50 113)(13 136 41 109)(14 82 42 115)(15 138 43 101)(16 84 44 117)(17 140 45 103)(18 86 46 119)(19 132 47 105)(20 88 48 111)(21 100 143 65)(22 58 144 79)(23 92 145 67)(24 60 146 71)(25 94 147 69)(26 52 148 73)(27 96 149 61)(28 54 150 75)(29 98 141 63)(30 56 142 77)(51 152 72 129)(53 154 74 121)(55 156 76 123)(57 158 78 125)(59 160 80 127)(62 122 97 155)(64 124 99 157)(66 126 91 159)(68 128 93 151)(70 130 95 153)
(1 70 12 52)(2 61 13 53)(3 62 14 54)(4 63 15 55)(5 64 16 56)(6 65 17 57)(7 66 18 58)(8 67 19 59)(9 68 20 60)(10 69 11 51)(21 140 158 118)(22 131 159 119)(23 132 160 120)(24 133 151 111)(25 134 152 112)(26 135 153 113)(27 136 154 114)(28 137 155 115)(29 138 156 116)(30 139 157 117)(31 93 48 71)(32 94 49 72)(33 95 50 73)(34 96 41 74)(35 97 42 75)(36 98 43 76)(37 99 44 77)(38 100 45 78)(39 91 46 79)(40 92 47 80)(81 149 109 121)(82 150 110 122)(83 141 101 123)(84 142 102 124)(85 143 103 125)(86 144 104 126)(87 145 105 127)(88 146 106 128)(89 147 107 129)(90 148 108 130)
(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 33 32)(2 31 34 9)(3 8 35 40)(4 39 36 7)(5 6 37 38)(11 50 49 12)(13 48 41 20)(14 19 42 47)(15 46 43 18)(16 17 44 45)(21 142 143 30)(22 29 144 141)(23 150 145 28)(24 27 146 149)(25 148 147 26)(51 73 72 52)(53 71 74 60)(54 59 75 80)(55 79 76 58)(56 57 77 78)(61 93 96 68)(62 67 97 92)(63 91 98 66)(64 65 99 100)(69 95 94 70)(81 111 114 88)(82 87 115 120)(83 119 116 86)(84 85 117 118)(89 113 112 90)(101 131 138 104)(102 103 139 140)(105 137 132 110)(106 109 133 136)(107 135 134 108)(121 151 154 128)(122 127 155 160)(123 159 156 126)(124 125 157 158)(129 153 152 130)```

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X 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 4 10 10 1 1 1 1 2 2 2 2 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 D4⋊2D5 D5×C4○D4 kernel C42.229D10 C4×Dic10 D5×C42 C23.D10 D10.12D4 C4.Dic10 D10⋊2Q8 C23.21D10 C4×C5⋊D4 C20.17D4 C20⋊2D4 D4×C20 C4×D4 C20 D10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C4 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 2 8 4 2 4 2 4 2 16 4 4

Matrix representation of C42.229D10 in GL4(𝔽41) generated by

 32 40 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 40 0 0 0 18 1 0 0 0 0 17 3 0 0 38 38
,
 40 9 0 0 18 1 0 0 0 0 17 1 0 0 38 24
`G:=sub<GL(4,GF(41))| [32,0,0,0,40,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[40,18,0,0,0,1,0,0,0,0,17,38,0,0,3,38],[40,18,0,0,9,1,0,0,0,0,17,38,0,0,1,24] >;`

C42.229D10 in GAP, Magma, Sage, TeX

`C_4^2._{229}D_{10}`
`% in TeX`

`G:=Group("C4^2.229D10");`
`// GroupNames label`

`G:=SmallGroup(320,1229);`
`// by ID`

`G=gap.SmallGroup(320,1229);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,794,12550]);`
`// 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,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

׿
×
𝔽