Copied to
clipboard

## G = C42.74D10order 320 = 26·5

### 74th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.74D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — C42.74D10
 Lower central C5 — C10 — C2×C20 — C42.74D10
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C42.74D10
G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1, dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 606 in 144 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4.4D4, C41D4, C2×D8, C2×SD16, C52C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C83D4, C2×C52C8, D10⋊C4, D4⋊D5, D4.D5, C4×C20, C2×Dic10, C2×D20, D4×C10, D4×C10, C42.D5, C4.D20, C2×D4⋊D5, C2×D4.D5, C5×C41D4, C42.74D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C41D4, C8⋊C22, C5⋊D4, C22×D5, C83D4, D4×D5, C2×C5⋊D4, D4.D10, C20⋊D4, C42.74D10

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 8 8 40 2 2 4 4 40 2 2 20 20 20 20 2 ··· 2 8 ··· 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C5⋊D4 C8⋊C22 D4×D5 D4.D10 kernel C42.74D10 C42.D5 C4.D20 C2×D4⋊D5 C2×D4.D5 C5×C4⋊1D4 C5⋊2C8 C2×C20 C4⋊1D4 C42 C2×D4 C2×C4 C10 C4 C2 # reps 1 1 1 2 2 1 4 2 2 2 4 8 2 4 8

Matrix representation of C42.74D10 in GL6(𝔽41)

 35 18 0 0 0 0 23 6 0 0 0 0 0 0 0 0 23 6 0 0 0 0 35 18 0 0 18 35 0 0 0 0 6 23 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0
,
 1 0 0 0 0 0 28 40 0 0 0 0 0 0 8 3 3 8 0 0 38 26 33 10 0 0 3 8 33 38 0 0 33 10 3 15
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 8 33 38 0 0 26 38 31 8 0 0 8 3 3 8 0 0 10 33 26 38

`G:=sub<GL(6,GF(41))| [35,23,0,0,0,0,18,6,0,0,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,23,35,0,0,0,0,6,18,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,8,38,3,33,0,0,3,26,8,10,0,0,3,33,33,3,0,0,8,10,38,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,26,8,10,0,0,8,38,3,33,0,0,33,31,3,26,0,0,38,8,8,38] >;`

C42.74D10 in GAP, Magma, Sage, TeX

`C_4^2._{74}D_{10}`
`% in TeX`

`G:=Group("C4^2.74D10");`
`// GroupNames label`

`G:=SmallGroup(320,701);`
`// by ID`

`G=gap.SmallGroup(320,701);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,1123,297,136,12550]);`
`// Polycyclic`

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

׿
×
𝔽