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## G = S3×D10⋊C4order 480 = 25·3·5

### Direct product of S3 and D10⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×D10⋊C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C22×S3×D5 — S3×D10⋊C4
 Lower central C15 — C30 — S3×D10⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for S3×D10⋊C4
G = < a,b,c,d,e | a3=b2=c10=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c5d >

Subgroups: 1612 in 264 conjugacy classes, 74 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C3, C4 [×4], C22, C22 [×22], C5, S3 [×4], S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], C23 [×11], D5 [×4], C10 [×3], C10 [×4], Dic3 [×2], C12 [×2], D6 [×6], D6 [×12], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×6], C4×S3 [×4], C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3 [×9], C22×C6, C5×S3 [×4], C3×D5 [×2], D15 [×2], C30 [×3], C2×C22⋊C4, C2×Dic5, C2×Dic5 [×3], C2×C20, C2×C20 [×3], C22×D5, C22×D5 [×9], C22×C10, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×6], D30 [×2], D30 [×2], C2×C30, D10⋊C4, D10⋊C4 [×3], C22×Dic5, C22×C20, C23×D5, S3×C22⋊C4, S3×Dic5 [×2], C6×Dic5, S3×C20 [×2], C10×Dic3, C2×Dic15, C2×C60, C2×S3×D5 [×4], C2×S3×D5 [×4], D5×C2×C6, S3×C2×C10, C22×D15, C2×D10⋊C4, D10⋊Dic3, D304C4, C3×D10⋊C4, D303C4, C2×S3×Dic5, S3×C2×C20, C22×S3×D5, S3×D10⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, S3×C2×C4, S3×D4 [×2], S3×D5, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, S3×C22⋊C4, C2×S3×D5, C2×D10⋊C4, C4×S3×D5, S3×D20, S3×C5⋊D4, S3×D10⋊C4

Smallest permutation representation of S3×D10⋊C4
On 120 points
Generators in S120
(1 23 40)(2 24 31)(3 25 32)(4 26 33)(5 27 34)(6 28 35)(7 29 36)(8 30 37)(9 21 38)(10 22 39)(11 112 110)(12 113 101)(13 114 102)(14 115 103)(15 116 104)(16 117 105)(17 118 106)(18 119 107)(19 120 108)(20 111 109)(41 66 53)(42 67 54)(43 68 55)(44 69 56)(45 70 57)(46 61 58)(47 62 59)(48 63 60)(49 64 51)(50 65 52)(71 96 83)(72 97 84)(73 98 85)(74 99 86)(75 100 87)(76 91 88)(77 92 89)(78 93 90)(79 94 81)(80 95 82)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 71)(18 72)(19 73)(20 74)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 61)(30 62)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)(99 109)(100 110)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 24)(22 23)(25 30)(26 29)(27 28)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)(71 73)(74 80)(75 79)(76 78)(81 87)(82 86)(83 85)(88 90)(91 93)(94 100)(95 99)(96 98)(101 103)(104 110)(105 109)(106 108)(111 117)(112 116)(113 115)(118 120)
(1 117 57 87)(2 118 58 88)(3 119 59 89)(4 120 60 90)(5 111 51 81)(6 112 52 82)(7 113 53 83)(8 114 54 84)(9 115 55 85)(10 116 56 86)(11 65 95 35)(12 66 96 36)(13 67 97 37)(14 68 98 38)(15 69 99 39)(16 70 100 40)(17 61 91 31)(18 62 92 32)(19 63 93 33)(20 64 94 34)(21 103 43 73)(22 104 44 74)(23 105 45 75)(24 106 46 76)(25 107 47 77)(26 108 48 78)(27 109 49 79)(28 110 50 80)(29 101 41 71)(30 102 42 72)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 15A 15B 20A ··· 20H 20I ··· 20P 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 3 3 3 3 10 10 30 30 2 2 2 6 6 10 10 30 30 2 2 2 2 2 20 20 2 ··· 2 6 ··· 6 4 4 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 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 C2 C2 C4 S3 D4 D5 D6 D6 D6 D10 D10 D10 C4×S3 C4×D5 D20 C5⋊D4 S3×D4 S3×D5 C2×S3×D5 C4×S3×D5 S3×D20 S3×C5⋊D4 kernel S3×D10⋊C4 D10⋊Dic3 D30⋊4C4 C3×D10⋊C4 D30⋊3C4 C2×S3×Dic5 S3×C2×C20 C22×S3×D5 C2×S3×D5 D10⋊C4 S3×C10 S3×C2×C4 C2×Dic5 C2×C20 C22×D5 C2×Dic3 C2×C12 C22×S3 D10 D6 D6 D6 C10 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 4 2 1 1 1 2 2 2 4 8 8 8 2 2 2 4 4 4

Matrix representation of S3×D10⋊C4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 60 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 1 1
,
 1 44 0 0 17 17 0 0 0 0 1 0 0 0 0 1
,
 60 17 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 32 7 0 0 54 29 0 0 0 0 50 0 0 0 0 50
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[1,0,0,0,0,1,0,0,0,0,60,1,0,0,0,1],[1,17,0,0,44,17,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,17,1,0,0,0,0,1,0,0,0,0,1],[32,54,0,0,7,29,0,0,0,0,50,0,0,0,0,50] >;

S3×D10⋊C4 in GAP, Magma, Sage, TeX

S_3\times D_{10}\rtimes C_4
% in TeX

G:=Group("S3xD10:C4");
// GroupNames label

G:=SmallGroup(480,548);
// by ID

G=gap.SmallGroup(480,548);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^10=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^5*d>;
// generators/relations

׿
×
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