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G = D70.C2order 280 = 23·5·7

The non-split extension by D70 of C2 acting faithfully

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C35 — D70.C2
C1C7C35C70C5×Dic7 — D70.C2
C35 — D70.C2
C1C2

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

35C2
35C2
5C4
7C4
35C22
7D5
7D5
5D7
5D7
35C2×C4
7C20
7D10
5C28
5D14
7C4×D5
5C4×D7

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 2A2B2C4A4B4C4D5A5B7A7B7C10A10B14A14B14C20A20B20C20D28A···28F35A···35F70A···70F
order122244445577710101414142020202028···2835···3570···70
size113535557722222222221414141410···104···44···4

40 irreducible representations

dim1111122222244
type++++++++++
imageC1C2C2C2C4D5D7D10D14C4×D5C4×D7D5×D7D70.C2
kernelD70.C2C7×Dic5C5×Dic7D70D35Dic7Dic5C14C10C7C5C2C1
# reps1111423234666

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

2803700
24424400
002277
002677
,
2803700
0100
0041280
00275240
,
280000
028000
0013815
00229143
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

Export

Subgroup lattice of D70.C2 in TeX

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