Copied to
clipboard

G = C4×D35order 280 = 23·5·7

Direct product of C4 and D35

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D35, C282D5, C202D7, C1402C2, C2.1D70, D70.2C2, C10.9D14, C14.9D10, Dic355C2, C70.9C22, C53(C4×D7), C72(C4×D5), C357(C2×C4), SmallGroup(280,25)

Series: Derived Chief Lower central Upper central

C1C35 — C4×D35
C1C7C35C70D70 — C4×D35
C35 — C4×D35
C1C4

Generators and relations for C4×D35
 G = < a,b,c | a4=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

35C2
35C2
35C4
35C22
7D5
7D5
5D7
5D7
35C2×C4
7Dic5
7D10
5D14
5Dic7
7C4×D5
5C4×D7

Smallest permutation representation of C4×D35
On 140 points
Generators in S140
(1 123 40 103)(2 124 41 104)(3 125 42 105)(4 126 43 71)(5 127 44 72)(6 128 45 73)(7 129 46 74)(8 130 47 75)(9 131 48 76)(10 132 49 77)(11 133 50 78)(12 134 51 79)(13 135 52 80)(14 136 53 81)(15 137 54 82)(16 138 55 83)(17 139 56 84)(18 140 57 85)(19 106 58 86)(20 107 59 87)(21 108 60 88)(22 109 61 89)(23 110 62 90)(24 111 63 91)(25 112 64 92)(26 113 65 93)(27 114 66 94)(28 115 67 95)(29 116 68 96)(30 117 69 97)(31 118 70 98)(32 119 36 99)(33 120 37 100)(34 121 38 101)(35 122 39 102)
(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 39)(2 38)(3 37)(4 36)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(71 119)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 126)(100 125)(101 124)(102 123)(103 122)(104 121)(105 120)

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

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

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

76 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B7A7B7C10A10B14A14B14C20A20B20C20D28A···28F35A···35L70A···70L140A···140X
order122244445577710101414142020202028···2835···3570···70140···140
size113535113535222222222222222···22···22···22···2

76 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4D5D7D10D14C4×D5C4×D7D35D70C4×D35
kernelC4×D35Dic35C140D70D35C28C20C14C10C7C5C4C2C1
# reps11114232346121224

Matrix representation of C4×D35 in GL3(𝔽281) generated by

22800
02800
00280
,
100
0243232
0494
,
100
0243232
010438
G:=sub<GL(3,GF(281))| [228,0,0,0,280,0,0,0,280],[1,0,0,0,243,49,0,232,4],[1,0,0,0,243,104,0,232,38] >;

C4×D35 in GAP, Magma, Sage, TeX

C_4\times D_{35}
% in TeX

G:=Group("C4xD35");
// GroupNames label

G:=SmallGroup(280,25);
// by ID

G=gap.SmallGroup(280,25);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,26,643,6004]);
// Polycyclic

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

Export

Subgroup lattice of C4×D35 in TeX

׿
×
𝔽