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## G = S3×C7×D5order 420 = 22·3·5·7

### Direct product of C7, S3 and D5

Aliases: S3×C7×D5, C356D6, D15⋊C14, C216D10, C1057C22, C15⋊(C2×C14), (C5×S3)⋊C14, C51(S3×C14), (C3×D5)⋊C14, C31(D5×C14), (S3×C35)⋊3C2, (D5×C21)⋊3C2, (C7×D15)⋊3C2, SmallGroup(420,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×C7×D5
 Chief series C1 — C5 — C15 — C105 — D5×C21 — S3×C7×D5
 Lower central C15 — S3×C7×D5
 Upper central C1 — C7

Generators and relations for S3×C7×D5
G = < a,b,c,d,e | a7=b3=c2=d5=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 S3×C7×D5
On 105 points
Generators in S105
(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 54 94)(2 55 95)(3 56 96)(4 50 97)(5 51 98)(6 52 92)(7 53 93)(8 39 58)(9 40 59)(10 41 60)(11 42 61)(12 36 62)(13 37 63)(14 38 57)(15 85 65)(16 86 66)(17 87 67)(18 88 68)(19 89 69)(20 90 70)(21 91 64)(22 100 43)(23 101 44)(24 102 45)(25 103 46)(26 104 47)(27 105 48)(28 99 49)(29 75 79)(30 76 80)(31 77 81)(32 71 82)(33 72 83)(34 73 84)(35 74 78)
(8 39)(9 40)(10 41)(11 42)(12 36)(13 37)(14 38)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 75)(30 76)(31 77)(32 71)(33 72)(34 73)(35 74)(50 97)(51 98)(52 92)(53 93)(54 94)(55 95)(56 96)
(1 70 79 102 59)(2 64 80 103 60)(3 65 81 104 61)(4 66 82 105 62)(5 67 83 99 63)(6 68 84 100 57)(7 69 78 101 58)(8 53 19 35 44)(9 54 20 29 45)(10 55 21 30 46)(11 56 15 31 47)(12 50 16 32 48)(13 51 17 33 49)(14 52 18 34 43)(22 38 92 88 73)(23 39 93 89 74)(24 40 94 90 75)(25 41 95 91 76)(26 42 96 85 77)(27 36 97 86 71)(28 37 98 87 72)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 53)(9 54)(10 55)(11 56)(12 50)(13 51)(14 52)(15 47)(16 48)(17 49)(18 43)(19 44)(20 45)(21 46)(22 88)(23 89)(24 90)(25 91)(26 85)(27 86)(28 87)(36 97)(37 98)(38 92)(39 93)(40 94)(41 95)(42 96)(64 103)(65 104)(66 105)(67 99)(68 100)(69 101)(70 102)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 6 7A ··· 7F 10A 10B 14A ··· 14F 14G ··· 14L 14M ··· 14R 15A 15B 21A ··· 21F 35A ··· 35L 42A ··· 42F 70A ··· 70L 105A ··· 105L order 1 2 2 2 3 5 5 6 7 ··· 7 10 10 14 ··· 14 14 ··· 14 14 ··· 14 15 15 21 ··· 21 35 ··· 35 42 ··· 42 70 ··· 70 105 ··· 105 size 1 3 5 15 2 2 2 10 1 ··· 1 6 6 3 ··· 3 5 ··· 5 15 ··· 15 4 4 2 ··· 2 2 ··· 2 10 ··· 10 6 ··· 6 4 ··· 4

84 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 C7 C14 C14 C14 S3 D5 D6 D10 S3×C7 C7×D5 S3×C14 D5×C14 S3×D5 S3×C7×D5 kernel S3×C7×D5 D5×C21 S3×C35 C7×D15 S3×D5 C5×S3 C3×D5 D15 C7×D5 S3×C7 C35 C21 D5 S3 C5 C3 C7 C1 # reps 1 1 1 1 6 6 6 6 1 2 1 2 6 12 6 12 2 12

Matrix representation of S3×C7×D5 in GL4(𝔽211) generated by

 123 0 0 0 0 123 0 0 0 0 58 0 0 0 0 58
,
 1 0 0 0 0 1 0 0 0 0 209 176 0 0 193 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 18 210
,
 0 1 0 0 210 178 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))| [123,0,0,0,0,123,0,0,0,0,58,0,0,0,0,58],[1,0,0,0,0,1,0,0,0,0,209,193,0,0,176,1],[1,0,0,0,0,1,0,0,0,0,1,18,0,0,0,210],[0,210,0,0,1,178,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3×C7×D5 in GAP, Magma, Sage, TeX

S_3\times C_7\times D_5
% in TeX

G:=Group("S3xC7xD5");
// GroupNames label

G:=SmallGroup(420,27);
// by ID

G=gap.SmallGroup(420,27);
# by ID

G:=PCGroup([5,-2,-2,-7,-3,-5,568,8404]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^3=c^2=d^5=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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