direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C15×C3⋊C8, C3⋊C120, C6.C60, C15⋊5C24, C32⋊3C40, C60.10C6, C12.2C30, C30.9C12, C60.16S3, C30.12Dic3, (C3×C15)⋊13C8, C20.4(C3×S3), C12.8(C5×S3), C4.2(S3×C15), (C3×C60).9C2, (C3×C6).2C20, C2.(Dic3×C15), (C3×C12).3C10, (C3×C30).10C4, C6.4(C5×Dic3), C10.3(C3×Dic3), SmallGroup(360,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C15×C3⋊C8 |
Generators and relations for C15×C3⋊C8
G = < a,b,c | a15=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C15×C3⋊C8
►On 120 points
Generators in S120
(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 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 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(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)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(1 75 46 78 116 40 19 95)(2 61 47 79 117 41 20 96)(3 62 48 80 118 42 21 97)(4 63 49 81 119 43 22 98)(5 64 50 82 120 44 23 99)(6 65 51 83 106 45 24 100)(7 66 52 84 107 31 25 101)(8 67 53 85 108 32 26 102)(9 68 54 86 109 33 27 103)(10 69 55 87 110 34 28 104)(11 70 56 88 111 35 29 105)(12 71 57 89 112 36 30 91)(13 72 58 90 113 37 16 92)(14 73 59 76 114 38 17 93)(15 74 60 77 115 39 18 94)
G:=sub<Sym(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), (1,11,6)(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)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(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)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115), (1,75,46,78,116,40,19,95)(2,61,47,79,117,41,20,96)(3,62,48,80,118,42,21,97)(4,63,49,81,119,43,22,98)(5,64,50,82,120,44,23,99)(6,65,51,83,106,45,24,100)(7,66,52,84,107,31,25,101)(8,67,53,85,108,32,26,102)(9,68,54,86,109,33,27,103)(10,69,55,87,110,34,28,104)(11,70,56,88,111,35,29,105)(12,71,57,89,112,36,30,91)(13,72,58,90,113,37,16,92)(14,73,59,76,114,38,17,93)(15,74,60,77,115,39,18,94)>;
G:=Group( (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), (1,11,6)(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)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(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)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115), (1,75,46,78,116,40,19,95)(2,61,47,79,117,41,20,96)(3,62,48,80,118,42,21,97)(4,63,49,81,119,43,22,98)(5,64,50,82,120,44,23,99)(6,65,51,83,106,45,24,100)(7,66,52,84,107,31,25,101)(8,67,53,85,108,32,26,102)(9,68,54,86,109,33,27,103)(10,69,55,87,110,34,28,104)(11,70,56,88,111,35,29,105)(12,71,57,89,112,36,30,91)(13,72,58,90,113,37,16,92)(14,73,59,76,114,38,17,93)(15,74,60,77,115,39,18,94) );
G=PermutationGroup([[(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)], [(1,11,6),(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),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(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),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115)], [(1,75,46,78,116,40,19,95),(2,61,47,79,117,41,20,96),(3,62,48,80,118,42,21,97),(4,63,49,81,119,43,22,98),(5,64,50,82,120,44,23,99),(6,65,51,83,106,45,24,100),(7,66,52,84,107,31,25,101),(8,67,53,85,108,32,26,102),(9,68,54,86,109,33,27,103),(10,69,55,87,110,34,28,104),(11,70,56,88,111,35,29,105),(12,71,57,89,112,36,30,91),(13,72,58,90,113,37,16,92),(14,73,59,76,114,38,17,93),(15,74,60,77,115,39,18,94)]])
180 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | ··· | 15H | 15I | ··· | 15T | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30H | 30I | ··· | 30T | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60AN | 120A | ··· | 120AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C8 | C10 | C12 | C15 | C20 | C24 | C30 | C40 | C60 | C120 | S3 | Dic3 | C3×S3 | C3⋊C8 | C5×S3 | C3×Dic3 | C5×Dic3 | C3×C3⋊C8 | S3×C15 | C5×C3⋊C8 | Dic3×C15 | C15×C3⋊C8 |
| kernel | C15×C3⋊C8 | C3×C60 | C5×C3⋊C8 | C3×C30 | C3×C3⋊C8 | C60 | C3×C15 | C3×C12 | C30 | C3⋊C8 | C3×C6 | C15 | C12 | C32 | C6 | C3 | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 | 32 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C15×C3⋊C8 ►in GL4(𝔽241) generated by
| 225 | 0 | 0 | 0 |
| 0 | 183 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 15 |
| 8 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [225,0,0,0,0,183,0,0,0,0,225,0,0,0,0,225],[1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,15],[8,0,0,0,0,64,0,0,0,0,0,1,0,0,1,0] >;
C15×C3⋊C8 in GAP, Magma, Sage, TeX
C_{15}\times C_3\rtimes C_8 % in TeX
G:=Group("C15xC3:C8"); // GroupNames label
G:=SmallGroup(360,34);
// by ID
G=gap.SmallGroup(360,34);
# by ID
G:=PCGroup([6,-2,-3,-5,-2,-2,-3,180,69,8645]);
// Polycyclic
G:=Group<a,b,c|a^15=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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