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## G = S3×D40order 480 = 25·3·5

### Direct product of S3 and D40

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — S3×D40
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — S3×D20 — S3×D40
 Lower central C15 — C30 — C60 — S3×D40
 Upper central C1 — C2 — C4 — C8

Generators and relations for S3×D40
G = < a,b,c,d | a3=b2=c40=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1308 in 152 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×2], S3 [×2], C6, C6 [×2], C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6, D6 [×6], C2×C6 [×2], C15, C2×C8, D8 [×4], C2×D4 [×2], C20, C20, D10 [×8], C2×C10, C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C2×D8, C40, C40, D20 [×2], D20 [×4], C2×C20, C22×D5 [×2], S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], C5×Dic3, C60, S3×D5 [×4], C6×D5 [×2], S3×C10, D30 [×2], D40, D40 [×3], C2×C40, C2×D20 [×2], S3×D8, C5×C3⋊C8, C120, C3⋊D20 [×2], C3×D20 [×2], S3×C20, D60 [×2], C2×S3×D5 [×2], C2×D40, C3⋊D40 [×2], C3×D40, S3×C40, D120, S3×D20 [×2], S3×D40
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C22×S3, C2×D8, D20 [×2], C22×D5, S3×D4, S3×D5, D40 [×2], C2×D20, S3×D8, C2×S3×D5, C2×D40, S3×D20, S3×D40

Smallest permutation representation of S3×D40
On 120 points
Generators in S120
(1 85 67)(2 86 68)(3 87 69)(4 88 70)(5 89 71)(6 90 72)(7 91 73)(8 92 74)(9 93 75)(10 94 76)(11 95 77)(12 96 78)(13 97 79)(14 98 80)(15 99 41)(16 100 42)(17 101 43)(18 102 44)(19 103 45)(20 104 46)(21 105 47)(22 106 48)(23 107 49)(24 108 50)(25 109 51)(26 110 52)(27 111 53)(28 112 54)(29 113 55)(30 114 56)(31 115 57)(32 116 58)(33 117 59)(34 118 60)(35 119 61)(36 120 62)(37 81 63)(38 82 64)(39 83 65)(40 84 66)
(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)(41 119)(42 120)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(81 103)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)

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

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

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

69 conjugacy classes

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

69 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D8 D10 D10 D10 D20 D20 D40 S3×D4 S3×D5 S3×D8 C2×S3×D5 S3×D20 S3×D40 kernel S3×D40 C3⋊D40 C3×D40 S3×C40 D120 S3×D20 D40 C5×Dic3 S3×C10 S3×C8 C40 D20 C5×S3 C3⋊C8 C24 C4×S3 Dic3 D6 S3 C10 C8 C5 C4 C2 C1 # reps 1 2 1 1 1 2 1 1 1 2 1 2 4 2 2 2 4 4 16 1 2 2 2 4 8

Matrix representation of S3×D40 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 197 163 0 0 0 0 78 78 0 0 0 0 0 0 195 70 0 0 0 0 208 24 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 190 240 0 0 0 0 0 0 47 147 0 0 0 0 44 194 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[197,78,0,0,0,0,163,78,0,0,0,0,0,0,195,208,0,0,0,0,70,24,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,190,0,0,0,0,0,240,0,0,0,0,0,0,47,44,0,0,0,0,147,194,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×D40 in GAP, Magma, Sage, TeX

S_3\times D_{40}
% in TeX

G:=Group("S3xD40");
// GroupNames label

G:=SmallGroup(480,328);
// by ID

G=gap.SmallGroup(480,328);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,142,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^40=d^2=1,b*a*b=a^-1,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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