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G = D4×C35order 280 = 23·5·7

Direct product of C35 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C35, C4⋊C70, C22⋊C70, C203C14, C283C10, C1407C2, C70.23C22, (C2×C70)⋊1C2, (C2×C14)⋊1C10, (C2×C10)⋊1C14, C2.1(C2×C70), C14.6(C2×C10), C10.6(C2×C14), SmallGroup(280,30)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C35
C1C2C14C70C2×C70 — D4×C35
C1C2 — D4×C35
C1C70 — D4×C35

Generators and relations for D4×C35
 G = < a,b,c | a35=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C10
2C10
2C14
2C14
2C70
2C70

Smallest permutation representation of D4×C35
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)
(1 111 102 63)(2 112 103 64)(3 113 104 65)(4 114 105 66)(5 115 71 67)(6 116 72 68)(7 117 73 69)(8 118 74 70)(9 119 75 36)(10 120 76 37)(11 121 77 38)(12 122 78 39)(13 123 79 40)(14 124 80 41)(15 125 81 42)(16 126 82 43)(17 127 83 44)(18 128 84 45)(19 129 85 46)(20 130 86 47)(21 131 87 48)(22 132 88 49)(23 133 89 50)(24 134 90 51)(25 135 91 52)(26 136 92 53)(27 137 93 54)(28 138 94 55)(29 139 95 56)(30 140 96 57)(31 106 97 58)(32 107 98 59)(33 108 99 60)(34 109 100 61)(35 110 101 62)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 106)(98 107)(99 108)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)

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

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

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

175 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D7A···7F10A10B10C10D10E···10L14A···14F14G···14R20A20B20C20D28A···28F35A···35X70A···70X70Y···70BT140A···140X
order1222455557···71010101010···1014···1414···142020202028···2835···3570···7070···70140···140
size1122211111···111112···21···12···222222···21···11···12···22···2

175 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C5C7C10C10C14C14C35C70C70D4C5×D4C7×D4D4×C35
kernelD4×C35C140C2×C70C7×D4C5×D4C28C2×C14C20C2×C10D4C4C22C35C7C5C1
# reps112464861224244814624

Matrix representation of D4×C35 in GL4(𝔽281) generated by

232000
07900
0010
0001
,
1000
0100
0038279
0020243
,
280000
028000
0038279
0019243
G:=sub<GL(4,GF(281))| [232,0,0,0,0,79,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,38,20,0,0,279,243],[280,0,0,0,0,280,0,0,0,0,38,19,0,0,279,243] >;

D4×C35 in GAP, Magma, Sage, TeX

D_4\times C_{35}
% in TeX

G:=Group("D4xC35");
// GroupNames label

G:=SmallGroup(280,30);
// by ID

G=gap.SmallGroup(280,30);
# by ID

G:=PCGroup([5,-2,-2,-5,-7,-2,1421]);
// Polycyclic

G:=Group<a,b,c|a^35=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D4×C35 in TeX

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