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## G = S3×Dic15order 360 = 23·32·5

### Direct product of S3 and Dic15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — S3×Dic15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3×Dic15 — S3×Dic15
 Lower central C3×C15 — S3×Dic15
 Upper central C1 — C2

Generators and relations for S3×Dic15
G = < a,b,c,d | a3=b2=c30=1, d2=c15, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 348 in 70 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C2×C10, C4×S3, C2×Dic3, C5×S3, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C2×Dic5, C3×C15, C3×Dic5, Dic15, Dic15, S3×C10, C2×C30, S3×Dic3, S3×C15, C3×C30, S3×Dic5, C2×Dic15, C3×Dic15, C3⋊Dic15, S3×C30, S3×Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, Dic5, D10, C4×S3, C2×Dic3, D15, S32, C2×Dic5, Dic15, S3×D5, D30, S3×Dic3, S3×Dic5, C2×Dic15, S3×D15, S3×Dic15

Smallest permutation representation of S3×Dic15
On 120 points
Generators in S120
(1 11 21)(2 12 22)(3 13 23)(4 14 24)(5 15 25)(6 16 26)(7 17 27)(8 18 28)(9 19 29)(10 20 30)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 101 111)(92 102 112)(93 103 113)(94 104 114)(95 105 115)(96 106 116)(97 107 117)(98 108 118)(99 109 119)(100 110 120)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 109)(70 110)(71 111)(72 112)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)
(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 61 16 76)(2 90 17 75)(3 89 18 74)(4 88 19 73)(5 87 20 72)(6 86 21 71)(7 85 22 70)(8 84 23 69)(9 83 24 68)(10 82 25 67)(11 81 26 66)(12 80 27 65)(13 79 28 64)(14 78 29 63)(15 77 30 62)(31 107 46 92)(32 106 47 91)(33 105 48 120)(34 104 49 119)(35 103 50 118)(36 102 51 117)(37 101 52 116)(38 100 53 115)(39 99 54 114)(40 98 55 113)(41 97 56 112)(42 96 57 111)(43 95 58 110)(44 94 59 109)(45 93 60 108)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 15C 15D 15E ··· 15J 30A 30B 30C 30D 30E ··· 30J 30K ··· 30R order 1 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 6 6 10 10 10 10 10 10 12 12 15 15 15 15 15 ··· 15 30 30 30 30 30 ··· 30 30 ··· 30 size 1 1 3 3 2 2 4 15 15 45 45 2 2 2 2 4 6 6 2 2 6 6 6 6 30 30 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + - + + - + + + - - + - image C1 C2 C2 C2 C4 S3 S3 D5 Dic3 D6 Dic5 D10 C4×S3 D15 Dic15 D30 S32 S3×D5 S3×Dic3 S3×Dic5 S3×D15 S3×Dic15 kernel S3×Dic15 C3×Dic15 C3⋊Dic15 S3×C30 S3×C15 Dic15 S3×C10 S3×C6 C5×S3 C30 C3×S3 C3×C6 C15 D6 S3 C6 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 4 1 1 2 2 2 4 2 2 4 8 4 1 2 1 2 4 4

Matrix representation of S3×Dic15 in GL6(𝔽61)

 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 60 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 60 60
,
 44 60 0 0 0 0 1 0 0 0 0 0 0 0 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 47 0 0 0 0 39 39 0 0 0 0 0 0 60 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[44,1,0,0,0,0,60,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,39,0,0,0,0,47,39,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×Dic15 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_{15}
% in TeX

G:=Group("S3xDic15");
// GroupNames label

G:=SmallGroup(360,78);
// by ID

G=gap.SmallGroup(360,78);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^30=1,d^2=c^15,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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