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## G = C3×D5×D7order 420 = 22·3·5·7

### Direct product of C3, D5 and D7

Aliases: C3×D5×D7, C215D10, C155D14, D353C6, C1055C22, C51(C6×D7), C74(C6×D5), C355(C2×C6), (C7×D5)⋊3C6, (C5×D7)⋊3C6, (C3×D35)⋊3C2, (D5×C21)⋊2C2, (D7×C15)⋊2C2, SmallGroup(420,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C3×D5×D7
 Chief series C1 — C7 — C35 — C105 — D7×C15 — C3×D5×D7
 Lower central C35 — C3×D5×D7
 Upper central C1 — C3

Generators and relations for C3×D5×D7
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 >

Smallest permutation representation of C3×D5×D7
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 2A 2B 2C 3A 3B 5A 5B 6A 6B 6C 6D 6E 6F 7A 7B 7C 10A 10B 14A 14B 14C 15A 15B 15C 15D 21A ··· 21F 30A 30B 30C 30D 35A ··· 35F 42A ··· 42F 105A ··· 105L order 1 2 2 2 3 3 5 5 6 6 6 6 6 6 7 7 7 10 10 14 14 14 15 15 15 15 21 ··· 21 30 30 30 30 35 ··· 35 42 ··· 42 105 ··· 105 size 1 5 7 35 1 1 2 2 5 5 7 7 35 35 2 2 2 14 14 10 10 10 2 2 2 2 2 ··· 2 14 14 14 14 4 ··· 4 10 ··· 10 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D5 D7 D10 D14 C3×D5 C3×D7 C6×D5 C6×D7 D5×D7 C3×D5×D7 kernel C3×D5×D7 D7×C15 D5×C21 C3×D35 D5×D7 C7×D5 C5×D7 D35 C3×D7 C3×D5 C21 C15 D7 D5 C7 C5 C3 C1 # reps 1 1 1 1 2 2 2 2 2 3 2 3 4 6 4 6 6 12

Matrix representation of C3×D5×D7 in GL4(𝔽211) generated by

 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 1 0 0 210 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 210 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
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] >;

C3×D5×D7 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

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