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G = C3xD35order 210 = 2·3·5·7

Direct product of C3 and D35

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

Aliases: C3xD35, C35:5C6, C15:2D7, C21:2D5, C105:2C2, C5:(C3xD7), C7:3(C3xD5), SmallGroup(210,7)

Series: Derived Chief Lower central Upper central

C1C35 — C3xD35
C1C7C35C105 — C3xD35
C35 — C3xD35
C1C3

Generators and relations for C3xD35
 G = < a,b,c | a3=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 104 in 16 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C3, C6, D5, D7, C3xD5, C3xD7, D35, C3xD35
35C2
35C6
7D5
5D7
7C3xD5
5C3xD7

Smallest permutation representation of C3xD35
On 105 points
Generators in S105
(1 91 47)(2 92 48)(3 93 49)(4 94 50)(5 95 51)(6 96 52)(7 97 53)(8 98 54)(9 99 55)(10 100 56)(11 101 57)(12 102 58)(13 103 59)(14 104 60)(15 105 61)(16 71 62)(17 72 63)(18 73 64)(19 74 65)(20 75 66)(21 76 67)(22 77 68)(23 78 69)(24 79 70)(25 80 36)(26 81 37)(27 82 38)(28 83 39)(29 84 40)(30 85 41)(31 86 42)(32 87 43)(33 88 44)(34 89 45)(35 90 46)
(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 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 75)(72 74)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)

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

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

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

C3xD35 is a maximal subgroup of   C3xD5xD7  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+++++
imageC1C2C3C6D5D7C3xD5C3xD7D35C3xD35
kernelC3xD35C105D35C35C21C15C7C5C3C1
# reps112223461224

Matrix representation of C3xD35 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] >;

C3xD35 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 C3xD35 in TeX

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