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## G = S3×D4×C10order 480 = 25·3·5

### Direct product of C10, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×D4×C10
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C22×C10 — S3×D4×C10
 Lower central C3 — C6 — S3×D4×C10
 Upper central C1 — C2×C10 — D4×C10

Generators and relations for S3×D4×C10
G = < a,b,c,d,e | a10=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1124 in 472 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C5×S3, C5×S3, C30, C30, C30, C22×D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C22×C10, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C22×C20, D4×C10, D4×C10, C23×C10, C2×S3×D4, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, D4×C15, S3×C2×C10, S3×C2×C10, S3×C2×C10, C22×C30, D4×C2×C10, S3×C2×C20, C10×D12, C5×S3×D4, C10×C3⋊D4, D4×C30, S3×C22×C10, S3×D4×C10
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C24, C2×C10, C22×S3, C5×S3, C22×D4, C5×D4, C22×C10, S3×D4, S3×C23, S3×C10, D4×C10, C23×C10, C2×S3×D4, S3×C2×C10, D4×C2×C10, C5×S3×D4, S3×C22×C10, S3×D4×C10

Smallest permutation representation of S3×D4×C10
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 86 97)(2 87 98)(3 88 99)(4 89 100)(5 90 91)(6 81 92)(7 82 93)(8 83 94)(9 84 95)(10 85 96)(11 42 33)(12 43 34)(13 44 35)(14 45 36)(15 46 37)(16 47 38)(17 48 39)(18 49 40)(19 50 31)(20 41 32)(21 102 75)(22 103 76)(23 104 77)(24 105 78)(25 106 79)(26 107 80)(27 108 71)(28 109 72)(29 110 73)(30 101 74)(51 69 112)(52 70 113)(53 61 114)(54 62 115)(55 63 116)(56 64 117)(57 65 118)(58 66 119)(59 67 120)(60 68 111)
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 111)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 71)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 108)(42 109)(43 110)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 97)(52 98)(53 99)(54 100)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 88)(62 89)(63 90)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)
(1 27 117 37)(2 28 118 38)(3 29 119 39)(4 30 120 40)(5 21 111 31)(6 22 112 32)(7 23 113 33)(8 24 114 34)(9 25 115 35)(10 26 116 36)(11 82 104 52)(12 83 105 53)(13 84 106 54)(14 85 107 55)(15 86 108 56)(16 87 109 57)(17 88 110 58)(18 89 101 59)(19 90 102 60)(20 81 103 51)(41 92 76 69)(42 93 77 70)(43 94 78 61)(44 95 79 62)(45 96 80 63)(46 97 71 64)(47 98 72 65)(48 99 73 66)(49 100 74 67)(50 91 75 68)
(1 37)(2 38)(3 39)(4 40)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 81)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 91)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 101)(60 102)(61 78)(62 79)(63 80)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 10A ··· 10L 10M ··· 10AB 10AC ··· 10AR 10AS ··· 10BH 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 3 3 3 3 6 6 6 6 2 2 2 6 6 1 1 1 1 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 4 4 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 4 ··· 4

150 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 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 S3 D4 D6 D6 D6 C5×S3 C5×D4 S3×C10 S3×C10 S3×C10 S3×D4 C5×S3×D4 kernel S3×D4×C10 S3×C2×C20 C10×D12 C5×S3×D4 C10×C3⋊D4 D4×C30 S3×C22×C10 C2×S3×D4 S3×C2×C4 C2×D12 S3×D4 C2×C3⋊D4 C6×D4 S3×C23 D4×C10 S3×C10 C2×C20 C5×D4 C22×C10 C2×D4 D6 C2×C4 D4 C23 C10 C2 # reps 1 1 1 8 2 1 2 4 4 4 32 8 4 8 1 4 1 4 2 4 16 4 16 8 2 8

Matrix representation of S3×D4×C10 in GL4(𝔽61) generated by

 41 0 0 0 0 41 0 0 0 0 1 0 0 0 0 1
,
 60 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 1 1 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 60 59 0 0 1 1
,
 60 0 0 0 0 60 0 0 0 0 60 59 0 0 0 1
G:=sub<GL(4,GF(61))| [41,0,0,0,0,41,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,59,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,59,1] >;

S3×D4×C10 in GAP, Magma, Sage, TeX

S_3\times D_4\times C_{10}
% in TeX

G:=Group("S3xD4xC10");
// GroupNames label

G:=SmallGroup(480,1154);
// by ID

G=gap.SmallGroup(480,1154);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,633,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^3=c^2=d^4=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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