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G = Dic35order 140 = 22·5·7

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C35 — Dic35
C1C7C35C70 — Dic35
C35 — Dic35
C1C2

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

35C4
7Dic5
5Dic7

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 4A4B5A5B7A7B7C10A10B14A14B14C35A···35L70A···70L
order124455777101014141435···3570···70
size11353522222222222···22···2

38 irreducible representations

dim111222222
type++++--+-
imageC1C2C4D5D7Dic5Dic7D35Dic35
kernelDic35C70C35C14C10C7C5C2C1
# reps11223231212

Matrix representation of Dic35 in GL3(𝔽281) generated by

28000
025148
0133113
,
22800
0256133
02725
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

Export

Subgroup lattice of Dic35 in TeX

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