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## G = C3×C5⋊D12order 360 = 23·32·5

### Direct product of C3 and C5⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3×C5⋊D12
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C32×Dic5 — C3×C5⋊D12
 Lower central C15 — C30 — C3×C5⋊D12
 Upper central C1 — C6

Generators and relations for C3×C5⋊D12
G = < a,b,c,d | a3=b5=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 300 in 70 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C10, C12, D6, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, D10, C2×C10, D12, C3×D4, C5×S3, C3×D5, D15, C30, C30, C3×C12, S3×C6, S3×C6, C5⋊D4, C3×C15, C3×Dic5, C3×Dic5, C6×D5, S3×C10, D30, C2×C30, C3×D12, S3×C15, C3×D15, C3×C30, C5⋊D12, C3×C5⋊D4, C32×Dic5, S3×C30, C6×D15, C3×C5⋊D12
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2×C6, C3×S3, D10, D12, C3×D4, C3×D5, S3×C6, C5⋊D4, S3×D5, C6×D5, C3×D12, C5⋊D12, C3×C5⋊D4, C3×S3×D5, C3×C5⋊D12

Smallest permutation representation of C3×C5⋊D12
On 120 points
Generators in S120
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)(97 105 101)(98 106 102)(99 107 103)(100 108 104)(109 117 113)(110 118 114)(111 119 115)(112 120 116)
(1 19 26 110 98)(2 99 111 27 20)(3 21 28 112 100)(4 101 113 29 22)(5 23 30 114 102)(6 103 115 31 24)(7 13 32 116 104)(8 105 117 33 14)(9 15 34 118 106)(10 107 119 35 16)(11 17 36 120 108)(12 97 109 25 18)(37 49 73 89 69)(38 70 90 74 50)(39 51 75 91 71)(40 72 92 76 52)(41 53 77 93 61)(42 62 94 78 54)(43 55 79 95 63)(44 64 96 80 56)(45 57 81 85 65)(46 66 86 82 58)(47 59 83 87 67)(48 68 88 84 60)
(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 49)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)(73 98)(74 97)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)

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

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

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

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D5 D6 C3×S3 D10 D12 C3×D4 C3×D5 S3×C6 C5⋊D4 C6×D5 C3×D12 C3×C5⋊D4 S3×D5 C5⋊D12 C3×S3×D5 C3×C5⋊D12 kernel C3×C5⋊D12 C32×Dic5 S3×C30 C6×D15 C5⋊D12 C3×Dic5 S3×C10 D30 C3×Dic5 C3×C15 S3×C6 C30 Dic5 C3×C6 C15 C15 D6 C10 C32 C6 C5 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 1 2 2 2 2 4 2 4 4 4 8 2 2 4 4

Matrix representation of C3×C5⋊D12 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 17 60 0 0 1 0
,
 32 0 0 0 0 21 0 0 0 0 2 59 0 0 32 59
,
 0 21 0 0 32 0 0 0 0 0 2 59 0 0 32 59
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,17,1,0,0,60,0],[32,0,0,0,0,21,0,0,0,0,2,32,0,0,59,59],[0,32,0,0,21,0,0,0,0,0,2,32,0,0,59,59] >;

C3×C5⋊D12 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes D_{12}
% in TeX

G:=Group("C3xC5:D12");
// GroupNames label

G:=SmallGroup(360,63);
// by ID

G=gap.SmallGroup(360,63);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,730,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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