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## G = Dic3×D15order 360 = 23·32·5

### Direct product of Dic3 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — Dic3×D15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C6×D15 — Dic3×D15
 Lower central C3×C15 — Dic3×D15
 Upper central C1 — C2

Generators and relations for Dic3×D15
G = < a,b,c,d | a6=c15=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 412 in 70 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C20, D10, C4×S3, C2×Dic3, C3×D5, D15, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C4×D5, C3×C15, C5×Dic3, Dic15, C60, C6×D5, D30, S3×Dic3, C3×D15, C3×C30, D5×Dic3, C4×D15, Dic3×C15, C3⋊Dic15, C6×D15, Dic3×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, D10, C4×S3, C2×Dic3, D15, S32, C4×D5, S3×D5, D30, S3×Dic3, D5×Dic3, C4×D15, S3×D15, Dic3×D15

Smallest permutation representation of Dic3×D15
On 120 points
Generators in S120
(1 25 11 20 6 30)(2 26 12 21 7 16)(3 27 13 22 8 17)(4 28 14 23 9 18)(5 29 15 24 10 19)(31 55 36 60 41 50)(32 56 37 46 42 51)(33 57 38 47 43 52)(34 58 39 48 44 53)(35 59 40 49 45 54)(61 79 66 84 71 89)(62 80 67 85 72 90)(63 81 68 86 73 76)(64 82 69 87 74 77)(65 83 70 88 75 78)(91 117 101 112 96 107)(92 118 102 113 97 108)(93 119 103 114 98 109)(94 120 104 115 99 110)(95 106 105 116 100 111)
(1 46 20 32)(2 47 21 33)(3 48 22 34)(4 49 23 35)(5 50 24 36)(6 51 25 37)(7 52 26 38)(8 53 27 39)(9 54 28 40)(10 55 29 41)(11 56 30 42)(12 57 16 43)(13 58 17 44)(14 59 18 45)(15 60 19 31)(61 116 84 95)(62 117 85 96)(63 118 86 97)(64 119 87 98)(65 120 88 99)(66 106 89 100)(67 107 90 101)(68 108 76 102)(69 109 77 103)(70 110 78 104)(71 111 79 105)(72 112 80 91)(73 113 81 92)(74 114 82 93)(75 115 83 94)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 75)(10 74)(11 73)(12 72)(13 71)(14 70)(15 69)(16 80)(17 79)(18 78)(19 77)(20 76)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 105)(45 104)(46 108)(47 107)(48 106)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 10A 10B 12A 12B 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 2 2 3 3 3 4 4 4 4 5 5 6 6 6 6 6 10 10 12 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 15 15 2 2 4 3 3 45 45 2 2 2 2 4 30 30 2 2 6 6 2 2 2 2 4 ··· 4 6 6 6 6 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 D10 C4×S3 D15 C4×D5 D30 C4×D15 S32 S3×D5 S3×Dic3 D5×Dic3 S3×D15 Dic3×D15 kernel Dic3×D15 Dic3×C15 C3⋊Dic15 C6×D15 C3×D15 C5×Dic3 D30 C3×Dic3 D15 C30 C3×C6 C15 Dic3 C32 C6 C3 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 2 4 4 4 8 1 2 1 2 4 4

Matrix representation of Dic3×D15 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 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
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 44 17 0 0 0 0 44 60 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
,
 60 44 0 0 0 0 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[44,44,0,0,0,0,17,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,44,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3×D15 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times D_{15}
% in TeX

G:=Group("Dic3xD15");
// GroupNames label

G:=SmallGroup(360,77);
// by ID

G=gap.SmallGroup(360,77);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,31,201,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^15=d^2=1,b^2=a^3,b*a*b^-1=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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