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## G = Dic35order 140 = 22·5·7

### Dicyclic group

Aliases: Dic35, C7⋊Dic5, C353C4, C10.D7, C2.D35, C14.D5, C52Dic7, C70.1C2, SmallGroup(140,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — Dic35
 Chief series C1 — C7 — C35 — C70 — Dic35
 Lower central C35 — Dic35
 Upper central C1 — C2

Generators and relations for Dic35
G = < a,b | a70=1, b2=a35, bab-1=a-1 >

Smallest permutation representation of Dic35
Regular action 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 87 36 122)(2 86 37 121)(3 85 38 120)(4 84 39 119)(5 83 40 118)(6 82 41 117)(7 81 42 116)(8 80 43 115)(9 79 44 114)(10 78 45 113)(11 77 46 112)(12 76 47 111)(13 75 48 110)(14 74 49 109)(15 73 50 108)(16 72 51 107)(17 71 52 106)(18 140 53 105)(19 139 54 104)(20 138 55 103)(21 137 56 102)(22 136 57 101)(23 135 58 100)(24 134 59 99)(25 133 60 98)(26 132 61 97)(27 131 62 96)(28 130 63 95)(29 129 64 94)(30 128 65 93)(31 127 66 92)(32 126 67 91)(33 125 68 90)(34 124 69 89)(35 123 70 88)

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

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

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

Dic35 is a maximal subgroup of
D7×Dic5  D5×Dic7  C35⋊D4  C35⋊Q8  Dic70  C4×D35  C357D4  C353C12  Dic105
Dic35 is a maximal quotient of
C353C8  Dic105

38 conjugacy classes

 class 1 2 4A 4B 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 35A ··· 35L 70A ··· 70L order 1 2 4 4 5 5 7 7 7 10 10 14 14 14 35 ··· 35 70 ··· 70 size 1 1 35 35 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

38 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C4 D5 D7 Dic5 Dic7 D35 Dic35 kernel Dic35 C70 C35 C14 C10 C7 C5 C2 C1 # reps 1 1 2 2 3 2 3 12 12

Matrix representation of Dic35 in GL3(𝔽281) generated by

 280 0 0 0 25 148 0 133 113
,
 228 0 0 0 256 133 0 27 25
G:=sub<GL(3,GF(281))| [280,0,0,0,25,133,0,148,113],[228,0,0,0,256,27,0,133,25] >;

Dic35 in GAP, Magma, Sage, TeX

{\rm Dic}_{35}
% in TeX

G:=Group("Dic35");
// GroupNames label

G:=SmallGroup(140,3);
// by ID

G=gap.SmallGroup(140,3);
# by ID

G:=PCGroup([4,-2,-2,-5,-7,8,194,1923]);
// Polycyclic

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

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