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G = C3×Dic30order 360 = 23·32·5

Direct product of C3 and Dic30

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic30, C60.1C6, C60.7S3, C157Dic6, C6.19D30, C30.51D6, C12.7D15, C326Dic10, Dic15.1C6, C4.(C3×D15), (C3×C15)⋊9Q8, C152(C3×Q8), C6.8(C6×D5), C52(C3×Dic6), C10.8(S3×C6), C20.1(C3×S3), C30.8(C2×C6), (C3×C60).2C2, C2.3(C6×D15), (C3×C12).2D5, C12.1(C3×D5), C32(C3×Dic10), (C3×C6).27D10, (C3×C30).37C22, (C3×Dic15).1C2, SmallGroup(360,100)

Series: Derived Chief Lower central Upper central

C1C30 — C3×Dic30
C1C5C15C30C3×C30C3×Dic15 — C3×Dic30
C15C30 — C3×Dic30
C1C6C12

Generators and relations for C3×Dic30
 G = < a,b,c | a3=b60=1, c2=b30, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 180 in 54 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C3 [×2], C3, C4, C4 [×2], C5, C6 [×2], C6, Q8, C32, C10, Dic3 [×2], C12 [×2], C12 [×3], C15 [×2], C15, C3×C6, Dic5 [×2], C20, Dic6, C3×Q8, C30 [×2], C30, C3×Dic3 [×2], C3×C12, Dic10, C3×C15, C3×Dic5 [×2], Dic15 [×2], C60 [×2], C60, C3×Dic6, C3×C30, C3×Dic10, Dic30, C3×Dic15 [×2], C3×C60, C3×Dic30
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], Q8, D5, D6, C2×C6, C3×S3, D10, Dic6, C3×Q8, C3×D5, D15, S3×C6, Dic10, C6×D5, D30, C3×Dic6, C3×D15, C3×Dic10, Dic30, C6×D15, C3×Dic30

Smallest permutation representation of C3×Dic30
On 120 points
Generators in S120
(1 21 41)(2 22 42)(3 23 43)(4 24 44)(5 25 45)(6 26 46)(7 27 47)(8 28 48)(9 29 49)(10 30 50)(11 31 51)(12 32 52)(13 33 53)(14 34 54)(15 35 55)(16 36 56)(17 37 57)(18 38 58)(19 39 59)(20 40 60)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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)
(1 116 31 86)(2 115 32 85)(3 114 33 84)(4 113 34 83)(5 112 35 82)(6 111 36 81)(7 110 37 80)(8 109 38 79)(9 108 39 78)(10 107 40 77)(11 106 41 76)(12 105 42 75)(13 104 43 74)(14 103 44 73)(15 102 45 72)(16 101 46 71)(17 100 47 70)(18 99 48 69)(19 98 49 68)(20 97 50 67)(21 96 51 66)(22 95 52 65)(23 94 53 64)(24 93 54 63)(25 92 55 62)(26 91 56 61)(27 90 57 120)(28 89 58 119)(29 88 59 118)(30 87 60 117)

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

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

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

99 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C5A5B6A6B6C6D6E10A10B12A···12H12I12J12K12L15A···15P20A20B20C20D30A···30P60A···60AF
order12333334445566666101012···121212121215···152020202030···3060···60
size1111222230302211222222···2303030302···222222···22···2

99 irreducible representations

dim11111122222222222222222222
type++++-+++-+-+-
imageC1C2C2C3C6C6S3Q8D5D6C3×S3D10Dic6C3×Q8C3×D5D15S3×C6Dic10C6×D5D30C3×Dic6C3×D15C3×Dic10Dic30C6×D15C3×Dic30
kernelC3×Dic30C3×Dic15C3×C60Dic30Dic15C60C60C3×C15C3×C12C30C20C3×C6C15C15C12C12C10C32C6C6C5C4C3C3C2C1
# reps121242112122224424444888816

Matrix representation of C3×Dic30 in GL2(𝔽61) generated by

470
047
,
180
3417
,
1123
050
G:=sub<GL(2,GF(61))| [47,0,0,47],[18,34,0,17],[11,0,23,50] >;

C3×Dic30 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{30}
% in TeX

G:=Group("C3xDic30");
// GroupNames label

G:=SmallGroup(360,100);
// by ID

G=gap.SmallGroup(360,100);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,79,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^60=1,c^2=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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