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## G = C42.139D10order 320 = 26·5

### 139th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.139D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.11D10 — C42.139D10
 Lower central C5 — C2×C10 — C42.139D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.139D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c-1 >

Subgroups: 614 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], C10 [×3], C10 [×2], C42, C42 [×6], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×7], C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4, C4.4D4, C422C2 [×4], C4⋊Q8, Dic10 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], C22.50C24, C4×Dic5 [×2], C4×Dic5 [×4], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C22×Dic5 [×2], D4×C10, Q8×C10, C4×Dic10 [×2], C23.11D10 [×2], Dic5.14D4 [×2], C23.D10 [×4], D4×Dic5, C20.17D4, Dic5⋊Q8, Q8×Dic5, C5×C4.4D4, C42.139D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.50C24, D42D5 [×2], C23×D5, C2×D42D5, D5×C4○D4, D4.10D10, C42.139D10

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

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 4P 4Q 4R 4S 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 2 2 2 2 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 2- 1+4 D4⋊2D5 D5×C4○D4 D4.10D10 kernel C42.139D10 C4×Dic10 C23.11D10 Dic5.14D4 C23.D10 D4×Dic5 C20.17D4 Dic5⋊Q8 Q8×Dic5 C5×C4.4D4 C4.4D4 Dic5 C20 C42 C22⋊C4 C2×D4 C2×Q8 C10 C4 C2 C2 # reps 1 2 2 2 4 1 1 1 1 1 2 4 4 2 8 2 2 1 4 4 4

Matrix representation of C42.139D10 in GL6(𝔽41)

 40 9 0 0 0 0 18 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 9 0 0 0 0 0 32
,
 1 0 0 0 0 0 23 40 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 39 40
,
 1 0 0 0 0 0 23 40 0 0 0 0 0 0 14 14 0 0 0 0 30 27 0 0 0 0 0 0 32 0 0 0 0 0 0 32

`G:=sub<GL(6,GF(41))| [40,18,0,0,0,0,9,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,9,32],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,39,0,0,0,0,0,40],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,14,30,0,0,0,0,14,27,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;`

C42.139D10 in GAP, Magma, Sage, TeX

`C_4^2._{139}D_{10}`
`% in TeX`

`G:=Group("C4^2.139D10");`
`// GroupNames label`

`G:=SmallGroup(320,1343);`
`// by ID`

`G=gap.SmallGroup(320,1343);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,794,297,12550]);`
`// Polycyclic`

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

׿
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