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## G = S3×Dic10order 240 = 24·3·5

### Direct product of S3 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×Dic10
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — S3×Dic10
 Lower central C15 — C30 — S3×Dic10
 Upper central C1 — C2 — C4

Generators and relations for S3×Dic10
G = < a,b,c,d | a3=b2=c20=1, d2=c10, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 280 in 76 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, Q8, C10, C10, Dic3, Dic3, C12, C12, D6, C15, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Dic6, C4×S3, C4×S3, C3×Q8, C5×S3, C30, Dic10, Dic10, C2×Dic5, C2×C20, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×Dic10, S3×Dic5, C15⋊Q8, C3×Dic10, S3×C20, Dic30, S3×Dic10
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, D10, C22×S3, Dic10, C22×D5, S3×Q8, S3×D5, C2×Dic10, C2×S3×D5, S3×Dic10

Smallest permutation representation of S3×Dic10
On 120 points
Generators in S120
(1 60 69)(2 41 70)(3 42 71)(4 43 72)(5 44 73)(6 45 74)(7 46 75)(8 47 76)(9 48 77)(10 49 78)(11 50 79)(12 51 80)(13 52 61)(14 53 62)(15 54 63)(16 55 64)(17 56 65)(18 57 66)(19 58 67)(20 59 68)(21 106 84)(22 107 85)(23 108 86)(24 109 87)(25 110 88)(26 111 89)(27 112 90)(28 113 91)(29 114 92)(30 115 93)(31 116 94)(32 117 95)(33 118 96)(34 119 97)(35 120 98)(36 101 99)(37 102 100)(38 103 81)(39 104 82)(40 105 83)
(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(81 103)(82 104)(83 105)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 101)(100 102)
(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 32 11 22)(2 31 12 21)(3 30 13 40)(4 29 14 39)(5 28 15 38)(6 27 16 37)(7 26 17 36)(8 25 18 35)(9 24 19 34)(10 23 20 33)(41 116 51 106)(42 115 52 105)(43 114 53 104)(44 113 54 103)(45 112 55 102)(46 111 56 101)(47 110 57 120)(48 109 58 119)(49 108 59 118)(50 107 60 117)(61 83 71 93)(62 82 72 92)(63 81 73 91)(64 100 74 90)(65 99 75 89)(66 98 76 88)(67 97 77 87)(68 96 78 86)(69 95 79 85)(70 94 80 84)

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

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

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

S3×Dic10 is a maximal subgroup of
Dic20⋊S3  C40.2D6  C60.10C23  Dic10.26D6  D20.39D6  C30.C24  C15⋊2- 1+4  D12.29D10  S3×Q8×D5
S3×Dic10 is a maximal quotient of
Dic35Dic10  Dic151Q8  Dic3⋊Dic10  Dic3014C4  Dic3.Dic10  Dic3.2Dic10  D6⋊Dic10  C60.45D4  C60.46D4  Dic3.3Dic10  C60.48D4  D61Dic10  D62Dic10  D63Dic10  D64Dic10  C204Dic6

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6 10A 10B 10C 10D 10E 10F 12A 12B 12C 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 60A 60B 60C 60D order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 10 10 10 10 10 10 12 12 12 15 15 20 20 20 20 20 20 20 20 30 30 60 60 60 60 size 1 1 3 3 2 2 6 10 10 30 30 2 2 2 2 2 6 6 6 6 4 20 20 4 4 2 2 2 2 6 6 6 6 4 4 4 4 4 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + + + - - + + - image C1 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D10 D10 D10 Dic10 S3×Q8 S3×D5 C2×S3×D5 S3×Dic10 kernel S3×Dic10 S3×Dic5 C15⋊Q8 C3×Dic10 S3×C20 Dic30 Dic10 C5×S3 C4×S3 Dic5 C20 Dic3 C12 D6 S3 C5 C4 C2 C1 # reps 1 2 2 1 1 1 1 2 2 2 1 2 2 2 8 1 2 2 4

Matrix representation of S3×Dic10 in GL4(𝔽61) generated by

 60 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 60 60 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 36 32 0 0 2 34
,
 60 0 0 0 0 60 0 0 0 0 27 31 0 0 4 34
G:=sub<GL(4,GF(61))| [60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,2,0,0,32,34],[60,0,0,0,0,60,0,0,0,0,27,4,0,0,31,34] >;

S3×Dic10 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{10}
% in TeX

G:=Group("S3xDic10");
// GroupNames label

G:=SmallGroup(240,128);
// by ID

G=gap.SmallGroup(240,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,218,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^20=1,d^2=c^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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