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G = (C3×C15)⋊9C8order 360 = 23·32·5

6th semidirect product of C3×C15 and C8 acting via C8/C2=C4

metabelian, soluble, monomial, A-group

Aliases: (C3×C15)⋊9C8, (C3×C6).Dic5, (C3×C30).6C4, C52(C322C8), C322(C52C8), C3⋊Dic3.1D5, C2.(C32⋊Dic5), C10.2(C32⋊C4), (C5×C3⋊Dic3).4C2, SmallGroup(360,56)

Series: Derived Chief Lower central Upper central

C1C3×C15 — (C3×C15)⋊9C8
C1C5C3×C15C3×C30C5×C3⋊Dic3 — (C3×C15)⋊9C8
C3×C15 — (C3×C15)⋊9C8
C1C2

Generators and relations for (C3×C15)⋊9C8
 G = < a,b,c | a3=b15=c8=1, ab=ba, cac-1=ab5, cbc-1=a-1b-1 >

2C3
2C3
9C4
2C6
2C6
2C15
2C15
45C8
6Dic3
6Dic3
9C20
2C30
2C30
9C52C8
6C5×Dic3
6C5×Dic3
5C322C8

Smallest permutation representation of (C3×C15)⋊9C8
On 120 points
Generators in S120
(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(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)
(1 114 57 89 20 105 34 70)(2 113 53 78 21 104 45 74)(3 112 49 82 22 103 41 63)(4 111 60 86 23 102 37 67)(5 110 56 90 24 101 33 71)(6 109 52 79 25 100 44 75)(7 108 48 83 26 99 40 64)(8 107 59 87 27 98 36 68)(9 106 55 76 28 97 32 72)(10 120 51 80 29 96 43 61)(11 119 47 84 30 95 39 65)(12 118 58 88 16 94 35 69)(13 117 54 77 17 93 31 73)(14 116 50 81 18 92 42 62)(15 115 46 85 19 91 38 66)

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

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

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

36 conjugacy classes

class 1  2 3A3B4A4B5A5B6A6B8A8B8C8D10A10B15A···15H20A20B20C20D30A···30H
order12334455668888101015···152020202030···30
size114499224445454545224···4181818184···4

36 irreducible representations

dim11112224444
type+++-+-
imageC1C2C4C8D5Dic5C52C8C32⋊C4C322C8C32⋊Dic5(C3×C15)⋊9C8
kernel(C3×C15)⋊9C8C5×C3⋊Dic3C3×C30C3×C15C3⋊Dic3C3×C6C32C10C5C2C1
# reps11242242288

Matrix representation of (C3×C15)⋊9C8 in GL6(𝔽241)

100000
010000
001000
000100
000001
0032225240240
,
240510000
1901900000
0091700
00665900
00216122098
002190143143
,
168820000
215730000
002341242040
00002401
008714317375
0010818017375

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,32,0,0,0,1,0,225,0,0,0,0,0,240,0,0,0,0,1,240],[240,190,0,0,0,0,51,190,0,0,0,0,0,0,91,66,216,219,0,0,7,59,122,0,0,0,0,0,0,143,0,0,0,0,98,143],[168,215,0,0,0,0,82,73,0,0,0,0,0,0,234,0,87,108,0,0,124,0,143,180,0,0,204,240,173,173,0,0,0,1,75,75] >;

(C3×C15)⋊9C8 in GAP, Magma, Sage, TeX

(C_3\times C_{15})\rtimes_9C_8
% in TeX

G:=Group("(C3xC15):9C8");
// GroupNames label

G:=SmallGroup(360,56);
// by ID

G=gap.SmallGroup(360,56);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-5,12,31,963,201,964,730,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^15=c^8=1,a*b=b*a,c*a*c^-1=a*b^5,c*b*c^-1=a^-1*b^-1>;
// generators/relations

Export

Subgroup lattice of (C3×C15)⋊9C8 in TeX

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