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## G = D70.C2order 280 = 23·5·7

### The non-split extension by D70 of C2 acting faithfully

Aliases: D70.C2, D352C4, Dic72D5, Dic52D7, C14.3D10, C10.3D14, C70.3C22, C71(C4×D5), C52(C4×D7), C356(C2×C4), C2.3(D5×D7), (C7×Dic5)⋊2C2, (C5×Dic7)⋊2C2, SmallGroup(280,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — D70.C2
 Chief series C1 — C7 — C35 — C70 — C5×Dic7 — D70.C2
 Lower central C35 — D70.C2
 Upper central C1 — C2

Generators and relations for D70.C2
G = < a,b,c | a70=b2=1, c2=a35, bab=a-1, cac-1=a41, cbc-1=a40b >

Smallest permutation representation of D70.C2
On 140 points
Generators in S140
```(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)
(1 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 112 36 77)(2 83 37 118)(3 124 38 89)(4 95 39 130)(5 136 40 101)(6 107 41 72)(7 78 42 113)(8 119 43 84)(9 90 44 125)(10 131 45 96)(11 102 46 137)(12 73 47 108)(13 114 48 79)(14 85 49 120)(15 126 50 91)(16 97 51 132)(17 138 52 103)(18 109 53 74)(19 80 54 115)(20 121 55 86)(21 92 56 127)(22 133 57 98)(23 104 58 139)(24 75 59 110)(25 116 60 81)(26 87 61 122)(27 128 62 93)(28 99 63 134)(29 140 64 105)(30 111 65 76)(31 82 66 117)(32 123 67 88)(33 94 68 129)(34 135 69 100)(35 106 70 71)```

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 20A 20B 20C 20D 28A ··· 28F 35A ··· 35F 70A ··· 70F order 1 2 2 2 4 4 4 4 5 5 7 7 7 10 10 14 14 14 20 20 20 20 28 ··· 28 35 ··· 35 70 ··· 70 size 1 1 35 35 5 5 7 7 2 2 2 2 2 2 2 2 2 2 14 14 14 14 10 ··· 10 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 D5 D7 D10 D14 C4×D5 C4×D7 D5×D7 D70.C2 kernel D70.C2 C7×Dic5 C5×Dic7 D70 D35 Dic7 Dic5 C14 C10 C7 C5 C2 C1 # reps 1 1 1 1 4 2 3 2 3 4 6 6 6

Matrix representation of D70.C2 in GL4(𝔽281) generated by

 280 37 0 0 244 244 0 0 0 0 227 7 0 0 267 7
,
 280 37 0 0 0 1 0 0 0 0 41 280 0 0 275 240
,
 280 0 0 0 0 280 0 0 0 0 138 15 0 0 229 143
`G:=sub<GL(4,GF(281))| [280,244,0,0,37,244,0,0,0,0,227,267,0,0,7,7],[280,0,0,0,37,1,0,0,0,0,41,275,0,0,280,240],[280,0,0,0,0,280,0,0,0,0,138,229,0,0,15,143] >;`

D70.C2 in GAP, Magma, Sage, TeX

`D_{70}.C_2`
`% in TeX`

`G:=Group("D70.C2");`
`// GroupNames label`

`G:=SmallGroup(280,9);`
`// by ID`

`G=gap.SmallGroup(280,9);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-7,20,26,328,6004]);`
`// Polycyclic`

`G:=Group<a,b,c|a^70=b^2=1,c^2=a^35,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^40*b>;`
`// generators/relations`

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