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## G = C42.158D10order 320 = 26·5

### 158th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.158D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊Q8 — C42.158D10
 Lower central C5 — C2×C10 — C42.158D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.158D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 950 in 220 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22.56C24, C10.D4, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4.D20, D10.13D4, C4⋊D20, D10⋊Q8, C5×C42.C2, C42.158D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.56C24, C23×D5, Q8.10D10, D48D10, C42.158D10

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D5 D10 D10 2+ 1+4 2- 1+4 Q8.10D10 D4⋊8D10 kernel C42.158D10 C4.D20 D10.13D4 C4⋊D20 D10⋊Q8 C5×C42.C2 C42.C2 C42 C4⋊C4 C10 C10 C2 C2 # reps 1 2 4 4 4 1 2 2 12 2 1 4 8

Matrix representation of C42.158D10 in GL8(𝔽41)

 11 9 0 0 0 0 0 0 32 30 0 0 0 0 0 0 0 0 11 9 0 0 0 0 0 0 32 30 0 0 0 0 0 0 0 0 39 13 20 26 0 0 0 0 28 2 21 5 0 0 0 0 2 6 11 28 0 0 0 0 8 8 22 30
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 28 13 0 0 0 0 0 40 28 0 0 0 0 0 0 38 1 0 0 0 0 0 3 38 0 1
,
 20 21 40 1 0 0 0 0 20 37 40 33 0 0 0 0 40 1 21 20 0 0 0 0 40 33 21 4 0 0 0 0 0 0 0 0 35 28 34 1 0 0 0 0 13 31 0 35 0 0 0 0 25 25 38 13 0 0 0 0 27 9 19 19
,
 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 1 0 13 28 0 0 0 0 35 40 32 37 0 0 0 0 0 0 7 35 0 0 0 0 0 0 8 34

`G:=sub<GL(8,GF(41))| [11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,39,28,2,8,0,0,0,0,13,2,6,8,0,0,0,0,20,21,11,22,0,0,0,0,26,5,28,30],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,3,0,0,0,0,0,40,38,38,0,0,0,0,28,28,1,0,0,0,0,0,13,0,0,1],[20,20,40,40,0,0,0,0,21,37,1,33,0,0,0,0,40,40,21,21,0,0,0,0,1,33,20,4,0,0,0,0,0,0,0,0,35,13,25,27,0,0,0,0,28,31,25,9,0,0,0,0,34,0,38,19,0,0,0,0,1,35,13,19],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,13,32,7,8,0,0,0,0,28,37,35,34] >;`

C42.158D10 in GAP, Magma, Sage, TeX

`C_4^2._{158}D_{10}`
`% in TeX`

`G:=Group("C4^2.158D10");`
`// GroupNames label`

`G:=SmallGroup(320,1372);`
`// by ID`

`G=gap.SmallGroup(320,1372);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,675,570,136,12550]);`
`// Polycyclic`

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

׿
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