Copied to
clipboard

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

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

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

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

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

׿
×
𝔽