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G = C3×C153C8order 360 = 23·32·5

Direct product of C3 and C153C8

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

Aliases: C3×C153C8, C153C24, C60.2C6, C30.5C12, C60.11S3, C12.8D15, C30.8Dic3, C6.4Dic15, C154(C3⋊C8), (C3×C15)⋊11C8, C6.(C3×Dic5), C20.2(C3×S3), (C3×C30).8C4, (C3×C60).4C2, C12.2(C3×D5), (C3×C12).3D5, C4.2(C3×D15), C2.(C3×Dic15), C323(C52C8), (C3×C6).2Dic5, C10.2(C3×Dic3), C52(C3×C3⋊C8), C3⋊(C3×C52C8), SmallGroup(360,35)

Series: Derived Chief Lower central Upper central

C1C15 — C3×C153C8
C1C5C15C30C60C3×C60 — C3×C153C8
C15 — C3×C153C8
C1C12

Generators and relations for C3×C153C8
 G = < a,b,c | a3=b15=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C15
15C8
2C12
2C30
5C3⋊C8
15C24
3C52C8
2C60
5C3×C3⋊C8
3C3×C52C8

Smallest permutation representation of C3×C153C8
On 120 points
Generators in S120
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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)
(1 110 47 84 19 92 45 73)(2 109 48 83 20 91 31 72)(3 108 49 82 21 105 32 71)(4 107 50 81 22 104 33 70)(5 106 51 80 23 103 34 69)(6 120 52 79 24 102 35 68)(7 119 53 78 25 101 36 67)(8 118 54 77 26 100 37 66)(9 117 55 76 27 99 38 65)(10 116 56 90 28 98 39 64)(11 115 57 89 29 97 40 63)(12 114 58 88 30 96 41 62)(13 113 59 87 16 95 42 61)(14 112 60 86 17 94 43 75)(15 111 46 85 18 93 44 74)

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

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

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

108 conjugacy classes

class 1  2 3A3B3C3D3E4A4B5A5B6A6B6C6D6E8A8B8C8D10A10B12A12B12C12D12E···12J15A···15P20A20B20C20D24A···24H30A···30P60A···60AF
order1233333445566666888810101212121212···1215···152020202024···2430···3060···60
size1111222112211222151515152211112···22···2222215···152···22···2

108 irreducible representations

dim11111111222222222222222222
type++++--+-
imageC1C2C3C4C6C8C12C24S3D5Dic3C3×S3Dic5C3⋊C8C3×D5D15C3×Dic3C52C8C3×Dic5Dic15C3×C3⋊C8C3×D15C3×C52C8C153C8C3×Dic15C3×C153C8
kernelC3×C153C8C3×C60C153C8C3×C30C60C3×C15C30C15C60C3×C12C30C20C3×C6C15C12C12C10C32C6C6C5C4C3C3C2C1
# reps112224481212224424444888816

Matrix representation of C3×C153C8 in GL5(𝔽241)

150000
015000
001500
00010
00001
,
10000
015000
0022500
000178190
00017911
,
2330000
00100
01000
00022491
0001817

G:=sub<GL(5,GF(241))| [15,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,15,0,0,0,0,0,225,0,0,0,0,0,178,179,0,0,0,190,11],[233,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,224,18,0,0,0,91,17] >;

C3×C153C8 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes_3C_8
% in TeX

G:=Group("C3xC15:3C8");
// GroupNames label

G:=SmallGroup(360,35);
// by ID

G=gap.SmallGroup(360,35);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-5,36,50,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^15=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×C153C8 in TeX

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