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## G = C5×S3×Dic3order 360 = 23·32·5

### Direct product of C5, S3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×S3×Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — S3×C30 — C5×S3×Dic3
 Lower central C32 — C5×S3×Dic3
 Upper central C1 — C10

Generators and relations for C5×S3×Dic3
G = < a,b,c,d,e | a5=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 164 in 70 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, C2×C10, C4×S3, C2×Dic3, C5×S3, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C2×C20, C3×C15, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, S3×Dic3, S3×C15, C3×C30, S3×C20, C10×Dic3, Dic3×C15, C5×C3⋊Dic3, S3×C30, C5×S3×Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C20, C2×C10, C4×S3, C2×Dic3, C5×S3, S32, C2×C20, C5×Dic3, S3×C10, S3×Dic3, S3×C20, C10×Dic3, C5×S32, C5×S3×Dic3

Smallest permutation representation of C5×S3×Dic3
On 120 points
Generators in S120
(1 25 19 13 7)(2 26 20 14 8)(3 27 21 15 9)(4 28 22 16 10)(5 29 23 17 11)(6 30 24 18 12)(31 55 49 43 37)(32 56 50 44 38)(33 57 51 45 39)(34 58 52 46 40)(35 59 53 47 41)(36 60 54 48 42)(61 85 79 73 67)(62 86 80 74 68)(63 87 81 75 69)(64 88 82 76 70)(65 89 83 77 71)(66 90 84 78 72)(91 115 109 103 97)(92 116 110 104 98)(93 117 111 105 99)(94 118 112 106 100)(95 119 113 107 101)(96 120 114 108 102)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 95 93)(92 96 94)(97 101 99)(98 102 100)(103 107 105)(104 108 106)(109 113 111)(110 114 112)(115 119 117)(116 120 118)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(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 32 4 35)(2 31 5 34)(3 36 6 33)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 50 22 53)(20 49 23 52)(21 54 24 51)(25 56 28 59)(26 55 29 58)(27 60 30 57)(61 92 64 95)(62 91 65 94)(63 96 66 93)(67 98 70 101)(68 97 71 100)(69 102 72 99)(73 104 76 107)(74 103 77 106)(75 108 78 105)(79 110 82 113)(80 109 83 112)(81 114 84 111)(85 116 88 119)(86 115 89 118)(87 120 90 117)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A ··· 15H 15I 15J 15K 15L 20A ··· 20H 20I ··· 20P 30A ··· 30H 30I 30J 30K 30L 30M ··· 30T 60A ··· 60H order 1 2 2 2 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 6 10 10 10 10 10 ··· 10 12 12 15 ··· 15 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 3 3 2 2 4 3 3 9 9 1 1 1 1 2 2 4 6 6 1 1 1 1 3 ··· 3 6 6 2 ··· 2 4 4 4 4 3 ··· 3 9 ··· 9 2 ··· 2 4 4 4 4 6 ··· 6 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 S3 Dic3 D6 C4×S3 C5×S3 C5×S3 C5×Dic3 S3×C10 S3×C20 S32 S3×Dic3 C5×S32 C5×S3×Dic3 kernel C5×S3×Dic3 Dic3×C15 C5×C3⋊Dic3 S3×C30 S3×C15 S3×Dic3 C3×Dic3 C3⋊Dic3 S3×C6 C3×S3 C5×Dic3 S3×C10 C5×S3 C30 C15 Dic3 D6 S3 C6 C3 C10 C5 C2 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 2 4 4 8 8 8 1 1 4 4

Matrix representation of C5×S3×Dic3 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 60 60 0 0 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 60 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 60 60 0 0 0 0 1 0
,
 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 1 0 0 0 0 0 60 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60] >;

C5×S3×Dic3 in GAP, Magma, Sage, TeX

C_5\times S_3\times {\rm Dic}_3
% in TeX

G:=Group("C5xS3xDic3");
// GroupNames label

G:=SmallGroup(360,72);
// by ID

G=gap.SmallGroup(360,72);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,127,1210,8645]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^6=1,e^2=d^3,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^-1=d^-1>;
// generators/relations

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