Copied to
clipboard

## G = C20.44D4order 160 = 25·5

### 1st non-split extension by C20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.44D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C20.44D4
 Lower central C5 — C10 — C20 — C20.44D4
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C20.44D4
G = < a,b,c | a20=b4=1, c2=a10, bab-1=cac-1=a-1, cbc-1=a5b-1 >

Smallest permutation representation of C20.44D4
Regular action on 160 points
Generators in S160
```(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)
(1 102 32 64)(2 101 33 63)(3 120 34 62)(4 119 35 61)(5 118 36 80)(6 117 37 79)(7 116 38 78)(8 115 39 77)(9 114 40 76)(10 113 21 75)(11 112 22 74)(12 111 23 73)(13 110 24 72)(14 109 25 71)(15 108 26 70)(16 107 27 69)(17 106 28 68)(18 105 29 67)(19 104 30 66)(20 103 31 65)(41 126 93 155)(42 125 94 154)(43 124 95 153)(44 123 96 152)(45 122 97 151)(46 121 98 150)(47 140 99 149)(48 139 100 148)(49 138 81 147)(50 137 82 146)(51 136 83 145)(52 135 84 144)(53 134 85 143)(54 133 86 142)(55 132 87 141)(56 131 88 160)(57 130 89 159)(58 129 90 158)(59 128 91 157)(60 127 92 156)
(1 131 11 121)(2 130 12 140)(3 129 13 139)(4 128 14 138)(5 127 15 137)(6 126 16 136)(7 125 17 135)(8 124 18 134)(9 123 19 133)(10 122 20 132)(21 151 31 141)(22 150 32 160)(23 149 33 159)(24 148 34 158)(25 147 35 157)(26 146 36 156)(27 145 37 155)(28 144 38 154)(29 143 39 153)(30 142 40 152)(41 102 51 112)(42 101 52 111)(43 120 53 110)(44 119 54 109)(45 118 55 108)(46 117 56 107)(47 116 57 106)(48 115 58 105)(49 114 59 104)(50 113 60 103)(61 86 71 96)(62 85 72 95)(63 84 73 94)(64 83 74 93)(65 82 75 92)(66 81 76 91)(67 100 77 90)(68 99 78 89)(69 98 79 88)(70 97 80 87)```

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 40A ··· 40P order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 20 20 20 20 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

46 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 Q16 D10 C4×D5 C5⋊D4 D20 C40⋊C2 Dic20 kernel C20.44D4 C4⋊Dic5 C2×C40 C2×Dic10 Dic10 C20 C2×C10 C2×C8 C10 C10 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 8 8

Matrix representation of C20.44D4 in GL5(𝔽41)

 40 0 0 0 0 0 6 40 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 23 32
,
 32 0 0 0 0 0 38 23 0 0 0 5 3 0 0 0 0 0 39 39 0 0 0 23 2
,
 1 0 0 0 0 0 38 23 0 0 0 5 3 0 0 0 0 0 35 35 0 0 0 13 6

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,1,0,0,0,40,0,0,0,0,0,0,9,23,0,0,0,0,32],[32,0,0,0,0,0,38,5,0,0,0,23,3,0,0,0,0,0,39,23,0,0,0,39,2],[1,0,0,0,0,0,38,5,0,0,0,23,3,0,0,0,0,0,35,13,0,0,0,35,6] >;`

C20.44D4 in GAP, Magma, Sage, TeX

`C_{20}._{44}D_4`
`% in TeX`

`G:=Group("C20.44D4");`
`// GroupNames label`

`G:=SmallGroup(160,23);`
`// by ID`

`G=gap.SmallGroup(160,23);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,73,79,362,86,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

Export

׿
×
𝔽