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## G = C5×D6⋊S3order 360 = 23·32·5

### Direct product of C5 and D6⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C5×D6⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — S3×C30 — C5×D6⋊S3
 Lower central C32 — C3×C6 — C5×D6⋊S3
 Upper central C1 — C10

Generators and relations for C5×D6⋊S3
G = < a,b,c,d,e | a5=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >

Subgroups: 196 in 70 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, C10, C10, Dic3, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, C2×C10, C3⋊D4, C5×S3, C30, C30, C3⋊Dic3, S3×C6, C5×D4, C3×C15, C5×Dic3, S3×C10, C2×C30, D6⋊S3, S3×C15, C3×C30, C5×C3⋊D4, C5×C3⋊Dic3, S3×C30, C5×D6⋊S3
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, C2×C10, C3⋊D4, C5×S3, S32, C5×D4, S3×C10, D6⋊S3, C5×C3⋊D4, C5×S32, C5×D6⋊S3

Smallest permutation representation of C5×D6⋊S3
On 120 points
Generators in S120
(1 29 23 17 11)(2 30 24 18 12)(3 25 19 13 7)(4 26 20 14 8)(5 27 21 15 9)(6 28 22 16 10)(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 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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 40)(8 39)(9 38)(10 37)(11 42)(12 41)(13 46)(14 45)(15 44)(16 43)(17 48)(18 47)(19 52)(20 51)(21 50)(22 49)(23 54)(24 53)(25 58)(26 57)(27 56)(28 55)(29 60)(30 59)(61 91)(62 96)(63 95)(64 94)(65 93)(66 92)(67 97)(68 102)(69 101)(70 100)(71 99)(72 98)(73 103)(74 108)(75 107)(76 106)(77 105)(78 104)(79 109)(80 114)(81 113)(82 112)(83 111)(84 110)(85 115)(86 120)(87 119)(88 118)(89 117)(90 116)
(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 65)(2 66)(3 61)(4 62)(5 63)(6 64)(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)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E ··· 10L 15A ··· 15H 15I 15J 15K 15L 20A 20B 20C 20D 30A ··· 30H 30I 30J 30K 30L 30M ··· 30AB order 1 2 2 2 3 3 3 4 5 5 5 5 6 6 6 6 6 6 6 10 10 10 10 10 ··· 10 15 ··· 15 15 15 15 15 20 20 20 20 30 ··· 30 30 30 30 30 30 ··· 30 size 1 1 6 6 2 2 4 18 1 1 1 1 2 2 4 6 6 6 6 1 1 1 1 6 ··· 6 2 ··· 2 4 4 4 4 18 18 18 18 2 ··· 2 4 4 4 4 6 ··· 6

75 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - image C1 C2 C2 C5 C10 C10 S3 D4 D6 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×C3⋊D4 S32 D6⋊S3 C5×S32 C5×D6⋊S3 kernel C5×D6⋊S3 C5×C3⋊Dic3 S3×C30 D6⋊S3 C3⋊Dic3 S3×C6 S3×C10 C3×C15 C30 C15 D6 C32 C6 C3 C10 C5 C2 C1 # reps 1 1 2 4 4 8 2 1 2 4 8 4 8 16 1 1 4 4

Matrix representation of C5×D6⋊S3 in GL6(𝔽61)

 34 0 0 0 0 0 0 34 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 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 0 1 0 0 0 0 60 60
,
 1 21 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 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 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
,
 31 30 0 0 0 0 29 30 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))| [34,0,0,0,0,0,0,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[1,0,0,0,0,0,21,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[31,29,0,0,0,0,30,30,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C5×D6⋊S3 in GAP, Magma, Sage, TeX

C_5\times D_6\rtimes S_3
% in TeX

G:=Group("C5xD6:S3");
// GroupNames label

G:=SmallGroup(360,74);
// by ID

G=gap.SmallGroup(360,74);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,1210,8645]);
// Polycyclic

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

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