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## G = S3×C2×C30order 360 = 23·32·5

### Direct product of C2×C30 and S3

Aliases: S3×C2×C30, C624C10, C6⋊(C2×C30), C304(C2×C6), (C2×C6)⋊5C30, C3⋊(C22×C30), (C2×C30)⋊11C6, (C6×C30)⋊10C2, C154(C22×C6), (C3×C30)⋊9C22, (C3×C15)⋊10C23, C322(C22×C10), (C3×C6)⋊2(C2×C10), SmallGroup(360,158)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C30
 Chief series C1 — C3 — C15 — C3×C15 — S3×C15 — S3×C30 — S3×C2×C30
 Lower central C3 — S3×C2×C30
 Upper central C1 — C2×C30

Generators and relations for S3×C2×C30
G = < a,b,c,d | a2=b30=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 236 in 138 conjugacy classes, 84 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C22, C22 [×6], C5, S3 [×4], C6 [×6], C6 [×7], C23, C32, C10 [×3], C10 [×4], D6 [×6], C2×C6 [×2], C2×C6 [×7], C15 [×2], C15, C3×S3 [×4], C3×C6 [×3], C2×C10, C2×C10 [×6], C22×S3, C22×C6, C5×S3 [×4], C30 [×6], C30 [×7], S3×C6 [×6], C62, C22×C10, C3×C15, S3×C10 [×6], C2×C30 [×2], C2×C30 [×7], S3×C2×C6, S3×C15 [×4], C3×C30 [×3], S3×C2×C10, C22×C30, S3×C30 [×6], C6×C30, S3×C2×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, S3, C6 [×7], C23, C10 [×7], D6 [×3], C2×C6 [×7], C15, C3×S3, C2×C10 [×7], C22×S3, C22×C6, C5×S3, C30 [×7], S3×C6 [×3], C22×C10, S3×C10 [×3], C2×C30 [×7], S3×C2×C6, S3×C15, S3×C2×C10, C22×C30, S3×C30 [×3], S3×C2×C30

Smallest permutation representation of S3×C2×C30
On 120 points
Generators in S120
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 71 81)(62 72 82)(63 73 83)(64 74 84)(65 75 85)(66 76 86)(67 77 87)(68 78 88)(69 79 89)(70 80 90)(91 101 111)(92 102 112)(93 103 113)(94 104 114)(95 105 115)(96 106 116)(97 107 117)(98 108 118)(99 109 119)(100 110 120)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)

G:=sub<Sym(120)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,71,81)(62,72,82)(63,73,83)(64,74,84)(65,75,85)(66,76,86)(67,77,87)(68,78,88)(69,79,89)(70,80,90)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)>;

G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,71,81)(62,72,82)(63,73,83)(64,74,84)(65,75,85)(66,76,86)(67,77,87)(68,78,88)(69,79,89)(70,80,90)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101) );

G=PermutationGroup([(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105)], [(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),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,21,11),(2,22,12),(3,23,13),(4,24,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,71,81),(62,72,82),(63,73,83),(64,74,84),(65,75,85),(66,76,86),(67,77,87),(68,78,88),(69,79,89),(70,80,90),(91,101,111),(92,102,112),(93,103,113),(94,104,114),(95,105,115),(96,106,116),(97,107,117),(98,108,118),(99,109,119),(100,110,120)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101)])

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A ··· 6F 6G ··· 6O 6P ··· 6W 10A ··· 10L 10M ··· 10AB 15A ··· 15H 15I ··· 15T 30A ··· 30X 30Y ··· 30BH 30BI ··· 30CN order 1 2 2 2 2 2 2 2 3 3 3 3 3 5 5 5 5 6 ··· 6 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 15 ··· 15 15 ··· 15 30 ··· 30 30 ··· 30 30 ··· 30 size 1 1 1 1 3 3 3 3 1 1 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 S3 D6 C3×S3 C5×S3 S3×C6 S3×C10 S3×C15 S3×C30 kernel S3×C2×C30 S3×C30 C6×C30 S3×C2×C10 S3×C2×C6 S3×C10 C2×C30 S3×C6 C62 C22×S3 D6 C2×C6 C2×C30 C30 C2×C10 C2×C6 C10 C6 C22 C2 # reps 1 6 1 2 4 12 2 24 4 8 48 8 1 3 2 4 6 12 8 24

Matrix representation of S3×C2×C30 in GL4(𝔽31) generated by

 1 0 0 0 0 30 0 0 0 0 1 0 0 0 0 1
,
 22 0 0 0 0 5 0 0 0 0 28 0 0 0 0 28
,
 1 0 0 0 0 1 0 0 0 0 5 0 0 0 0 25
,
 30 0 0 0 0 30 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(31))| [1,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,5,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,25],[30,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0] >;

S3×C2×C30 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{30}
% in TeX

G:=Group("S3xC2xC30");
// GroupNames label

G:=SmallGroup(360,158);
// by ID

G=gap.SmallGroup(360,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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