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G = C3xD5xD7order 420 = 22·3·5·7

Direct product of C3, D5 and D7

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

Aliases: C3xD5xD7, C21:5D10, C15:5D14, D35:3C6, C105:5C22, C5:1(C6xD7), C7:4(C6xD5), C35:5(C2xC6), (C7xD5):3C6, (C5xD7):3C6, (C3xD35):3C2, (D5xC21):2C2, (D7xC15):2C2, SmallGroup(420,24)

Series: Derived Chief Lower central Upper central

C1C35 — C3xD5xD7
C1C7C35C105D7xC15 — C3xD5xD7
C35 — C3xD5xD7
C1C3

Generators and relations for C3xD5xD7
 G = < a,b,c,d,e | a3=b5=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 256 in 40 conjugacy classes, 20 normal (all characteristic)
Quotients: C1, C2, C3, C22, C6, D5, C2xC6, D7, D10, D14, C3xD5, C3xD7, C6xD5, C6xD7, D5xD7, C3xD5xD7
5C2
7C2
35C2
35C22
5C6
7C6
35C6
7C10
7D5
5D7
5C14
35C2xC6
7D10
5D14
7C3xD5
7C30
5C3xD7
5C42
7C6xD5
5C6xD7

Smallest permutation representation of C3xD5xD7
On 105 points
Generators in S105
(1 76 41)(2 77 42)(3 71 36)(4 72 37)(5 73 38)(6 74 39)(7 75 40)(8 78 43)(9 79 44)(10 80 45)(11 81 46)(12 82 47)(13 83 48)(14 84 49)(15 85 50)(16 86 51)(17 87 52)(18 88 53)(19 89 54)(20 90 55)(21 91 56)(22 92 57)(23 93 58)(24 94 59)(25 95 60)(26 96 61)(27 97 62)(28 98 63)(29 99 64)(30 100 65)(31 101 66)(32 102 67)(33 103 68)(34 104 69)(35 105 70)
(1 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)(71 99 92 85 78)(72 100 93 86 79)(73 101 94 87 80)(74 102 95 88 81)(75 103 96 89 82)(76 104 97 90 83)(77 105 98 91 84)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)

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

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

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

60 conjugacy classes

class 1 2A2B2C3A3B5A5B6A6B6C6D6E6F7A7B7C10A10B14A14B14C15A15B15C15D21A···21F30A30B30C30D35A···35F42A···42F105A···105L
order1222335566666677710101414141515151521···213030303035···3542···42105···105
size15735112255773535222141410101022222···2141414144···410···104···4

60 irreducible representations

dim111111112222222244
type+++++++++
imageC1C2C2C2C3C6C6C6D5D7D10D14C3xD5C3xD7C6xD5C6xD7D5xD7C3xD5xD7
kernelC3xD5xD7D7xC15D5xC21C3xD35D5xD7C7xD5C5xD7D35C3xD7C3xD5C21C15D7D5C7C5C3C1
# reps1111222223234646612

Matrix representation of C3xD5xD7 in GL4(F211) generated by

1000
0100
001960
000196
,
1000
0100
00321
002100
,
1000
0100
0001
0010
,
0100
2101800
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(211))| [1,0,0,0,0,1,0,0,0,0,196,0,0,0,0,196],[1,0,0,0,0,1,0,0,0,0,32,210,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,210,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C3xD5xD7 in GAP, Magma, Sage, TeX

C_3\times D_5\times D_7
% in TeX

G:=Group("C3xD5xD7");
// GroupNames label

G:=SmallGroup(420,24);
// by ID

G=gap.SmallGroup(420,24);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,488,9004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^5=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of C3xD5xD7 in TeX

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