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G = C2xC6xD15order 360 = 23·32·5

Direct product of C2xC6 and D15

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2xC6xD15, C30:7D6, C62:4D5, C6:2(C6xD5), C10:2(S3xC6), (C6xC30):5C2, C30:2(C2xC6), (C2xC30):7C6, (C2xC30):9S3, (C3xC6):5D10, (C3xC15):8C23, C15:8(C22xS3), C15:2(C22xC6), (C3xC30):7C22, C32:6(C22xD5), C5:2(S3xC2xC6), C3:2(D5xC2xC6), (C2xC6):5(C3xD5), (C2xC10):7(C3xS3), SmallGroup(360,159)

Series: Derived Chief Lower central Upper central

C1C15 — C2xC6xD15
C1C5C15C3xC15C3xD15C6xD15 — C2xC6xD15
C15 — C2xC6xD15
C1C2xC6

Generators and relations for C2xC6xD15
 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, C2xC6, C2xC6, C15, C15, C3xS3, C3xC6, D10, C2xC10, C22xS3, C22xC6, C3xD5, D15, C30, C30, S3xC6, C62, C22xD5, C3xC15, C6xD5, D30, C2xC30, C2xC30, S3xC2xC6, C3xD15, C3xC30, D5xC2xC6, C22xD15, C6xD15, C6xC30, C2xC6xD15
Quotients: C1, C2, C3, C22, S3, C6, C23, D5, D6, C2xC6, C3xS3, D10, C22xS3, C22xC6, C3xD5, D15, S3xC6, C22xD5, C6xD5, D30, S3xC2xC6, C3xD15, D5xC2xC6, C22xD15, C6xD15, C2xC6xD15

Smallest permutation representation of C2xC6xD15
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 2A2B2C2D2E2F2G3A3B3C3D3E5A5B6A···6F6G···6O6P···6W10A···10F15A···15P30A···30AV
order1222222233333556···66···66···610···1015···1530···30
size11111515151511222221···12···215···152···22···22···2

108 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D5D6C3xS3D10C3xD5D15S3xC6C6xD5D30C3xD15C6xD15
kernelC2xC6xD15C6xD15C6xC30C22xD15D30C2xC30C2xC30C62C30C2xC10C3xC6C2xC6C2xC6C10C6C6C22C2
# reps1612122123264461212824

Matrix representation of C2xC6xD15 in GL4(F31) generated by

1000
0100
00300
00030
,
30000
03000
0050
0005
,
301800
131300
002020
00014
,
272200
12400
001720
002914
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] >;

C2xC6xD15 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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