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## G = S3×C40order 240 = 24·3·5

### Direct product of C40 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C40
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — S3×C40
 Lower central C3 — S3×C40
 Upper central C1 — C40

Generators and relations for S3×C40
G = < a,b,c | a40=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C40
On 120 points
Generators in S120
(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 94 43)(2 95 44)(3 96 45)(4 97 46)(5 98 47)(6 99 48)(7 100 49)(8 101 50)(9 102 51)(10 103 52)(11 104 53)(12 105 54)(13 106 55)(14 107 56)(15 108 57)(16 109 58)(17 110 59)(18 111 60)(19 112 61)(20 113 62)(21 114 63)(22 115 64)(23 116 65)(24 117 66)(25 118 67)(26 119 68)(27 120 69)(28 81 70)(29 82 71)(30 83 72)(31 84 73)(32 85 74)(33 86 75)(34 87 76)(35 88 77)(36 89 78)(37 90 79)(38 91 80)(39 92 41)(40 93 42)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)

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

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

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

S3×C40 is a maximal subgroup of   C40.52D6  C40.54D6  C40.55D6  D6.1D20  D407S3  D1205C2

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 5C 5D 6 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40P 40Q ··· 40AF 60A ··· 60H 120A ··· 120P order 1 2 2 2 3 4 4 4 4 5 5 5 5 6 8 8 8 8 8 8 8 8 10 10 10 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 24 24 24 30 30 30 30 40 ··· 40 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 3 3 2 1 1 3 3 1 1 1 1 2 1 1 1 1 3 3 3 3 1 1 1 1 3 ··· 3 2 2 2 2 2 2 1 ··· 1 3 ··· 3 2 2 2 2 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 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 C2 C4 C4 C5 C8 C10 C10 C10 C20 C20 C40 S3 D6 C4×S3 C5×S3 S3×C8 S3×C10 S3×C20 S3×C40 kernel S3×C40 C5×C3⋊C8 C120 S3×C20 C5×Dic3 S3×C10 S3×C8 C5×S3 C3⋊C8 C24 C4×S3 Dic3 D6 S3 C40 C20 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 2 2 4 8 4 4 4 8 8 32 1 1 2 4 4 4 8 16

Matrix representation of S3×C40 in GL2(𝔽41) generated by

 6 0 0 6
,
 40 34 6 0
,
 40 34 0 1
G:=sub<GL(2,GF(41))| [6,0,0,6],[40,6,34,0],[40,0,34,1] >;

S3×C40 in GAP, Magma, Sage, TeX

S_3\times C_{40}
% in TeX

G:=Group("S3xC40");
// GroupNames label

G:=SmallGroup(240,49);
// by ID

G=gap.SmallGroup(240,49);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,127,69,5765]);
// Polycyclic

G:=Group<a,b,c|a^40=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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