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G = C3×C15⋊C8order 360 = 23·32·5

Direct product of C3 and C15⋊C8

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

Aliases: C3×C15⋊C8, C151C24, C30.1C12, C30.4Dic3, C6.(C3×F5), (C3×C15)⋊6C8, C152(C3⋊C8), C323(C5⋊C8), C6.4(C3⋊F5), (C3×C6).2F5, (C3×C30).3C4, C10.(C3×Dic3), (C3×Dic5).5C6, (C3×Dic5).9S3, Dic5.2(C3×S3), (C32×Dic5).5C2, C5⋊(C3×C3⋊C8), C3⋊(C3×C5⋊C8), C2.(C3×C3⋊F5), SmallGroup(360,53)

Series: Derived Chief Lower central Upper central

C1C15 — C3×C15⋊C8
C1C5C15C30C3×Dic5C32×Dic5 — C3×C15⋊C8
C15 — C3×C15⋊C8
C1C6

Generators and relations for C3×C15⋊C8
 G = < a,b,c | a3=b15=c8=1, ab=ba, ac=ca, cbc-1=b2 >

2C3
5C4
2C6
2C15
15C8
5C12
5C12
10C12
2C30
5C3⋊C8
15C24
5C3×C12
3C5⋊C8
2C3×Dic5
5C3×C3⋊C8
3C3×C5⋊C8

Smallest permutation representation of C3×C15⋊C8
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)
(1 120 54 76 21 105 41 71)(2 113 58 78 22 98 45 73)(3 106 47 80 23 91 34 75)(4 114 51 82 24 99 38 62)(5 107 55 84 25 92 42 64)(6 115 59 86 26 100 31 66)(7 108 48 88 27 93 35 68)(8 116 52 90 28 101 39 70)(9 109 56 77 29 94 43 72)(10 117 60 79 30 102 32 74)(11 110 49 81 16 95 36 61)(12 118 53 83 17 103 40 63)(13 111 57 85 18 96 44 65)(14 119 46 87 19 104 33 67)(15 112 50 89 20 97 37 69)

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

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

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

54 conjugacy classes

class 1  2 3A3B3C3D3E4A4B 5 6A6B6C6D6E8A8B8C8D 10 12A12B12C12D12E···12J15A···15H24A···24H30A···30H
order1233333445666668888101212121212···1215···1524···2430···30
size111122255411222151515154555510···104···415···154···4

54 irreducible representations

dim1111111122222244444444
type+++-+-
imageC1C2C3C4C6C8C12C24S3Dic3C3×S3C3⋊C8C3×Dic3C3×C3⋊C8F5C5⋊C8C3×F5C3⋊F5C3×C5⋊C8C15⋊C8C3×C3⋊F5C3×C15⋊C8
kernelC3×C15⋊C8C32×Dic5C15⋊C8C3×C30C3×Dic5C3×C15C30C15C3×Dic5C30Dic5C15C10C5C3×C6C32C6C6C3C3C2C1
# reps1122244811222411222244

Matrix representation of C3×C15⋊C8 in GL6(𝔽241)

100000
010000
00225000
00022500
00002250
00000225
,
22500000
0150000
0009300
0013210900
000057226
000072226
,
0300000
21100000
000010
000001
0018723400
001415400

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[225,0,0,0,0,0,0,15,0,0,0,0,0,0,0,132,0,0,0,0,93,109,0,0,0,0,0,0,57,72,0,0,0,0,226,226],[0,211,0,0,0,0,30,0,0,0,0,0,0,0,0,0,187,141,0,0,0,0,234,54,0,0,1,0,0,0,0,0,0,1,0,0] >;

C3×C15⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_{15}\rtimes C_8
% in TeX

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

G:=SmallGroup(360,53);
// by ID

G=gap.SmallGroup(360,53);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,-5,36,50,1444,7781,1745]);
// 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^2>;
// generators/relations

Export

Subgroup lattice of C3×C15⋊C8 in TeX

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