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## G = C5×S3×D7order 420 = 22·3·5·7

### Direct product of C5, S3 and D7

Aliases: C5×S3×D7, C355D6, D21⋊C10, C156D14, C1056C22, C21⋊(C2×C10), (S3×C7)⋊C10, C71(S3×C10), (C3×D7)⋊C10, C31(C10×D7), (S3×C35)⋊2C2, (C5×D21)⋊3C2, (D7×C15)⋊3C2, SmallGroup(420,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C5×S3×D7
 Chief series C1 — C7 — C21 — C105 — D7×C15 — C5×S3×D7
 Lower central C21 — C5×S3×D7
 Upper central C1 — C5

Generators and relations for C5×S3×D7
G = < a,b,c,d,e | a5=b3=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 C5×S3×D7
On 105 points
Generators in S105
(1 90 69 48 27)(2 91 70 49 28)(3 85 64 43 22)(4 86 65 44 23)(5 87 66 45 24)(6 88 67 46 25)(7 89 68 47 26)(8 92 71 50 29)(9 93 72 51 30)(10 94 73 52 31)(11 95 74 53 32)(12 96 75 54 33)(13 97 76 55 34)(14 98 77 56 35)(15 99 78 57 36)(16 100 79 58 37)(17 101 80 59 38)(18 102 81 60 39)(19 103 82 61 40)(20 104 83 62 41)(21 105 84 63 42)
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(43 50 57)(44 51 58)(45 52 59)(46 53 60)(47 54 61)(48 55 62)(49 56 63)(64 71 78)(65 72 79)(66 73 80)(67 74 81)(68 75 82)(69 76 83)(70 77 84)(85 92 99)(86 93 100)(87 94 101)(88 95 102)(89 96 103)(90 97 104)(91 98 105)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(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)| (1,90,69,48,27)(2,91,70,49,28)(3,85,64,43,22)(4,86,65,44,23)(5,87,66,45,24)(6,88,67,46,25)(7,89,68,47,26)(8,92,71,50,29)(9,93,72,51,30)(10,94,73,52,31)(11,95,74,53,32)(12,96,75,54,33)(13,97,76,55,34)(14,98,77,56,35)(15,99,78,57,36)(16,100,79,58,37)(17,101,80,59,38)(18,102,81,60,39)(19,103,82,61,40)(20,104,83,62,41)(21,105,84,63,42), (1,13,20)(2,14,21)(3,8,15)(4,9,16)(5,10,17)(6,11,18)(7,12,19)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,50,57)(44,51,58)(45,52,59)(46,53,60)(47,54,61)(48,55,62)(49,56,63)(64,71,78)(65,72,79)(66,73,80)(67,74,81)(68,75,82)(69,76,83)(70,77,84)(85,92,99)(86,93,100)(87,94,101)(88,95,102)(89,96,103)(90,97,104)(91,98,105), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (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( (1,90,69,48,27)(2,91,70,49,28)(3,85,64,43,22)(4,86,65,44,23)(5,87,66,45,24)(6,88,67,46,25)(7,89,68,47,26)(8,92,71,50,29)(9,93,72,51,30)(10,94,73,52,31)(11,95,74,53,32)(12,96,75,54,33)(13,97,76,55,34)(14,98,77,56,35)(15,99,78,57,36)(16,100,79,58,37)(17,101,80,59,38)(18,102,81,60,39)(19,103,82,61,40)(20,104,83,62,41)(21,105,84,63,42), (1,13,20)(2,14,21)(3,8,15)(4,9,16)(5,10,17)(6,11,18)(7,12,19)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,50,57)(44,51,58)(45,52,59)(46,53,60)(47,54,61)(48,55,62)(49,56,63)(64,71,78)(65,72,79)(66,73,80)(67,74,81)(68,75,82)(69,76,83)(70,77,84)(85,92,99)(86,93,100)(87,94,101)(88,95,102)(89,96,103)(90,97,104)(91,98,105), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (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([[(1,90,69,48,27),(2,91,70,49,28),(3,85,64,43,22),(4,86,65,44,23),(5,87,66,45,24),(6,88,67,46,25),(7,89,68,47,26),(8,92,71,50,29),(9,93,72,51,30),(10,94,73,52,31),(11,95,74,53,32),(12,96,75,54,33),(13,97,76,55,34),(14,98,77,56,35),(15,99,78,57,36),(16,100,79,58,37),(17,101,80,59,38),(18,102,81,60,39),(19,103,82,61,40),(20,104,83,62,41),(21,105,84,63,42)], [(1,13,20),(2,14,21),(3,8,15),(4,9,16),(5,10,17),(6,11,18),(7,12,19),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42),(43,50,57),(44,51,58),(45,52,59),(46,53,60),(47,54,61),(48,55,62),(49,56,63),(64,71,78),(65,72,79),(66,73,80),(67,74,81),(68,75,82),(69,76,83),(70,77,84),(85,92,99),(86,93,100),(87,94,101),(88,95,102),(89,96,103),(90,97,104),(91,98,105)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(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)]])

75 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 5C 5D 6 7A 7B 7C 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 14A 14B 14C 15A 15B 15C 15D 21A 21B 21C 30A 30B 30C 30D 35A ··· 35L 70A ··· 70L 105A ··· 105L order 1 2 2 2 3 5 5 5 5 6 7 7 7 10 10 10 10 10 10 10 10 10 10 10 10 14 14 14 15 15 15 15 21 21 21 30 30 30 30 35 ··· 35 70 ··· 70 105 ··· 105 size 1 3 7 21 2 1 1 1 1 14 2 2 2 3 3 3 3 7 7 7 7 21 21 21 21 6 6 6 2 2 2 2 4 4 4 14 14 14 14 2 ··· 2 6 ··· 6 4 ··· 4

75 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 C5 C10 C10 C10 S3 D6 D7 D14 C5×S3 S3×C10 C5×D7 C10×D7 S3×D7 C5×S3×D7 kernel C5×S3×D7 D7×C15 S3×C35 C5×D21 S3×D7 S3×C7 C3×D7 D21 C5×D7 C35 C5×S3 C15 D7 C7 S3 C3 C5 C1 # reps 1 1 1 1 4 4 4 4 1 1 3 3 4 4 12 12 3 12

Matrix representation of C5×S3×D7 in GL5(𝔽211)

 107 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 210 210 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 210 0 0 0 0 0 1 0 0 0 0 210 210 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 193 1 0 0 0 209 129
,
 210 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 129 210 0 0 0 182 82

G:=sub<GL(5,GF(211))| [107,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,210,1,0,0,0,210,0,0,0,0,0,0,1,0,0,0,0,0,1],[210,0,0,0,0,0,1,210,0,0,0,0,210,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,193,209,0,0,0,1,129],[210,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,129,182,0,0,0,210,82] >;

C5×S3×D7 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_7
% in TeX

G:=Group("C5xS3xD7");
// GroupNames label

G:=SmallGroup(420,25);
// by ID

G=gap.SmallGroup(420,25);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-7,408,9004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=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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