metabelian, soluble, monomial, A-group
Aliases: (C3×C15)⋊9C8, (C3×C6).Dic5, (C3×C30).6C4, C5⋊2(C32⋊2C8), C32⋊2(C5⋊2C8), C3⋊Dic3.1D5, C2.(C32⋊Dic5), C10.2(C32⋊C4), (C5×C3⋊Dic3).4C2, SmallGroup(360,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5 — C3×C15 — C3×C30 — C5×C3⋊Dic3 — (C3×C15)⋊9C8 |
| C3×C15 — (C3×C15)⋊9C8 |
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 >
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 | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 4 | 4 | 9 | 9 | 2 | 2 | 4 | 4 | 45 | 45 | 45 | 45 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | D5 | Dic5 | C5⋊2C8 | C32⋊C4 | C32⋊2C8 | C32⋊Dic5 | (C3×C15)⋊9C8 |
| kernel | (C3×C15)⋊9C8 | C5×C3⋊Dic3 | C3×C30 | C3×C15 | C3⋊Dic3 | C3×C6 | C32 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of (C3×C15)⋊9C8 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 32 | 225 | 240 | 240 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 7 | 0 | 0 |
| 0 | 0 | 66 | 59 | 0 | 0 |
| 0 | 0 | 216 | 122 | 0 | 98 |
| 0 | 0 | 219 | 0 | 143 | 143 |
| 168 | 82 | 0 | 0 | 0 | 0 |
| 215 | 73 | 0 | 0 | 0 | 0 |
| 0 | 0 | 234 | 124 | 204 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 87 | 143 | 173 | 75 |
| 0 | 0 | 108 | 180 | 173 | 75 |
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
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