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G = C3×D35order 210 = 2·3·5·7

Direct product of C3 and D35

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

Aliases: C3×D35, C355C6, C152D7, C212D5, C1052C2, C5⋊(C3×D7), C73(C3×D5), SmallGroup(210,7)

Series: Derived Chief Lower central Upper central

C1C35 — C3×D35
C1C7C35C105 — C3×D35
C35 — C3×D35
C1C3

Generators and relations for C3×D35
 G = < a,b,c | a3=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

35C2
35C6
7D5
5D7
7C3×D5
5C3×D7

Smallest permutation representation of C3×D35
On 105 points
Generators in S105
(1 88 70)(2 89 36)(3 90 37)(4 91 38)(5 92 39)(6 93 40)(7 94 41)(8 95 42)(9 96 43)(10 97 44)(11 98 45)(12 99 46)(13 100 47)(14 101 48)(15 102 49)(16 103 50)(17 104 51)(18 105 52)(19 71 53)(20 72 54)(21 73 55)(22 74 56)(23 75 57)(24 76 58)(25 77 59)(26 78 60)(27 79 61)(28 80 62)(29 81 63)(30 82 64)(31 83 65)(32 84 66)(33 85 67)(34 86 68)(35 87 69)
(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)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 68)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(69 70)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)

G:=sub<Sym(105)| (1,88,70)(2,89,36)(3,90,37)(4,91,38)(5,92,39)(6,93,40)(7,94,41)(8,95,42)(9,96,43)(10,97,44)(11,98,45)(12,99,46)(13,100,47)(14,101,48)(15,102,49)(16,103,50)(17,104,51)(18,105,52)(19,71,53)(20,72,54)(21,73,55)(22,74,56)(23,75,57)(24,76,58)(25,77,59)(26,78,60)(27,79,61)(28,80,62)(29,81,63)(30,82,64)(31,83,65)(32,84,66)(33,85,67)(34,86,68)(35,87,69), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(69,70)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)>;

G:=Group( (1,88,70)(2,89,36)(3,90,37)(4,91,38)(5,92,39)(6,93,40)(7,94,41)(8,95,42)(9,96,43)(10,97,44)(11,98,45)(12,99,46)(13,100,47)(14,101,48)(15,102,49)(16,103,50)(17,104,51)(18,105,52)(19,71,53)(20,72,54)(21,73,55)(22,74,56)(23,75,57)(24,76,58)(25,77,59)(26,78,60)(27,79,61)(28,80,62)(29,81,63)(30,82,64)(31,83,65)(32,84,66)(33,85,67)(34,86,68)(35,87,69), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,68)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(69,70)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88) );

G=PermutationGroup([(1,88,70),(2,89,36),(3,90,37),(4,91,38),(5,92,39),(6,93,40),(7,94,41),(8,95,42),(9,96,43),(10,97,44),(11,98,45),(12,99,46),(13,100,47),(14,101,48),(15,102,49),(16,103,50),(17,104,51),(18,105,52),(19,71,53),(20,72,54),(21,73,55),(22,74,56),(23,75,57),(24,76,58),(25,77,59),(26,78,60),(27,79,61),(28,80,62),(29,81,63),(30,82,64),(31,83,65),(32,84,66),(33,85,67),(34,86,68),(35,87,69)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(36,68),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(69,70),(71,104),(72,103),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88)])

C3×D35 is a maximal subgroup of   C3×D5×D7  D15⋊D7

57 conjugacy classes

class 1  2 3A3B5A5B6A6B7A7B7C15A15B15C15D21A···21F35A···35L105A···105X
order123355667771515151521···2135···35105···105
size1351122353522222222···22···22···2

57 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6D5D7C3×D5C3×D7D35C3×D35
kernelC3×D35C105D35C35C21C15C7C5C3C1
# reps112223461224

Matrix representation of C3×D35 in GL2(𝔽211) generated by

1960
0196
,
127186
50148
,
187137
17624
G:=sub<GL(2,GF(211))| [196,0,0,196],[127,50,186,148],[187,176,137,24] >;

C3×D35 in GAP, Magma, Sage, TeX

C_3\times D_{35}
% in TeX

G:=Group("C3xD35");
// GroupNames label

G:=SmallGroup(210,7);
// by ID

G=gap.SmallGroup(210,7);
# by ID

G:=PCGroup([4,-2,-3,-5,-7,290,2883]);
// Polycyclic

G:=Group<a,b,c|a^3=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D35 in TeX

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