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## G = C2×C6×D15order 360 = 23·32·5

### Direct product of C2×C6 and D15

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C6×D15
G = < a,b,c,d | a2=b6=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 588 in 138 conjugacy classes, 62 normal (18 characteristic)
C1, C2, C2, C3, C3, C22, C22, C5, S3, C6, C6, C23, C32, D5, C10, D6, C2×C6, C2×C6, C15, C15, C3×S3, C3×C6, D10, C2×C10, C22×S3, C22×C6, C3×D5, D15, C30, C30, S3×C6, C62, C22×D5, C3×C15, C6×D5, D30, C2×C30, C2×C30, S3×C2×C6, C3×D15, C3×C30, D5×C2×C6, C22×D15, C6×D15, C6×C30, C2×C6×D15
Quotients: C1, C2, C3, C22, S3, C6, C23, D5, D6, C2×C6, C3×S3, D10, C22×S3, C22×C6, C3×D5, D15, S3×C6, C22×D5, C6×D5, D30, S3×C2×C6, C3×D15, D5×C2×C6, C22×D15, C6×D15, C2×C6×D15

Smallest permutation representation of C2×C6×D15
On 120 points
Generators in S120
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 45)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(61 118)(62 119)(63 120)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 105)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)
(1 27 6 17 11 22)(2 28 7 18 12 23)(3 29 8 19 13 24)(4 30 9 20 14 25)(5 16 10 21 15 26)(31 49 36 54 41 59)(32 50 37 55 42 60)(33 51 38 56 43 46)(34 52 39 57 44 47)(35 53 40 58 45 48)(61 81 71 76 66 86)(62 82 72 77 67 87)(63 83 73 78 68 88)(64 84 74 79 69 89)(65 85 75 80 70 90)(91 109 101 119 96 114)(92 110 102 120 97 115)(93 111 103 106 98 116)(94 112 104 107 99 117)(95 113 105 108 100 118)
(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 82)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 109)(32 108)(33 107)(34 106)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 110)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 105)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 5A 5B 6A ··· 6F 6G ··· 6O 6P ··· 6W 10A ··· 10F 15A ··· 15P 30A ··· 30AV order 1 2 2 2 2 2 2 2 3 3 3 3 3 5 5 6 ··· 6 6 ··· 6 6 ··· 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 1 1 15 15 15 15 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 15 ··· 15 2 ··· 2 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D5 D6 C3×S3 D10 C3×D5 D15 S3×C6 C6×D5 D30 C3×D15 C6×D15 kernel C2×C6×D15 C6×D15 C6×C30 C22×D15 D30 C2×C30 C2×C30 C62 C30 C2×C10 C3×C6 C2×C6 C2×C6 C10 C6 C6 C22 C2 # reps 1 6 1 2 12 2 1 2 3 2 6 4 4 6 12 12 8 24

Matrix representation of C2×C6×D15 in GL4(𝔽31) generated by

 1 0 0 0 0 1 0 0 0 0 30 0 0 0 0 30
,
 30 0 0 0 0 30 0 0 0 0 5 0 0 0 0 5
,
 30 18 0 0 13 13 0 0 0 0 20 20 0 0 0 14
,
 27 22 0 0 12 4 0 0 0 0 17 20 0 0 29 14
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[30,0,0,0,0,30,0,0,0,0,5,0,0,0,0,5],[30,13,0,0,18,13,0,0,0,0,20,0,0,0,20,14],[27,12,0,0,22,4,0,0,0,0,17,29,0,0,20,14] >;

C2×C6×D15 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{15}
% in TeX

G:=Group("C2xC6xD15");
// GroupNames label

G:=SmallGroup(360,159);
// by ID

G=gap.SmallGroup(360,159);
# by ID

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

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