direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C15⋊C8, C15⋊1C24, C30.1C12, C30.4Dic3, C6.(C3×F5), (C3×C15)⋊6C8, C15⋊2(C3⋊C8), C32⋊3(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
| C15 — C3×C15⋊C8 |
Generators and relations for C3×C15⋊C8
G = < a,b,c | a3=b15=c8=1, ab=ba, ac=ca, cbc-1=b2 >
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)
(1 106 46 76 24 97 38 61)(2 114 50 78 25 105 42 63)(3 107 54 80 26 98 31 65)(4 115 58 82 27 91 35 67)(5 108 47 84 28 99 39 69)(6 116 51 86 29 92 43 71)(7 109 55 88 30 100 32 73)(8 117 59 90 16 93 36 75)(9 110 48 77 17 101 40 62)(10 118 52 79 18 94 44 64)(11 111 56 81 19 102 33 66)(12 119 60 83 20 95 37 68)(13 112 49 85 21 103 41 70)(14 120 53 87 22 96 45 72)(15 113 57 89 23 104 34 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), (1,106,46,76,24,97,38,61)(2,114,50,78,25,105,42,63)(3,107,54,80,26,98,31,65)(4,115,58,82,27,91,35,67)(5,108,47,84,28,99,39,69)(6,116,51,86,29,92,43,71)(7,109,55,88,30,100,32,73)(8,117,59,90,16,93,36,75)(9,110,48,77,17,101,40,62)(10,118,52,79,18,94,44,64)(11,111,56,81,19,102,33,66)(12,119,60,83,20,95,37,68)(13,112,49,85,21,103,41,70)(14,120,53,87,22,96,45,72)(15,113,57,89,23,104,34,74)>;
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), (1,106,46,76,24,97,38,61)(2,114,50,78,25,105,42,63)(3,107,54,80,26,98,31,65)(4,115,58,82,27,91,35,67)(5,108,47,84,28,99,39,69)(6,116,51,86,29,92,43,71)(7,109,55,88,30,100,32,73)(8,117,59,90,16,93,36,75)(9,110,48,77,17,101,40,62)(10,118,52,79,18,94,44,64)(11,111,56,81,19,102,33,66)(12,119,60,83,20,95,37,68)(13,112,49,85,21,103,41,70)(14,120,53,87,22,96,45,72)(15,113,57,89,23,104,34,74) );
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)], [(1,106,46,76,24,97,38,61),(2,114,50,78,25,105,42,63),(3,107,54,80,26,98,31,65),(4,115,58,82,27,91,35,67),(5,108,47,84,28,99,39,69),(6,116,51,86,29,92,43,71),(7,109,55,88,30,100,32,73),(8,117,59,90,16,93,36,75),(9,110,48,77,17,101,40,62),(10,118,52,79,18,94,44,64),(11,111,56,81,19,102,33,66),(12,119,60,83,20,95,37,68),(13,112,49,85,21,103,41,70),(14,120,53,87,22,96,45,72),(15,113,57,89,23,104,34,74)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 5 | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | ··· | 15H | 24A | ··· | 24H | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 24 | ··· | 24 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 4 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | ··· | 4 | 15 | ··· | 15 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3×C3⋊C8 | F5 | C5⋊C8 | C3×F5 | C3⋊F5 | C3×C5⋊C8 | C15⋊C8 | C3×C3⋊F5 | C3×C15⋊C8 |
| kernel | C3×C15⋊C8 | C32×Dic5 | C15⋊C8 | C3×C30 | C3×Dic5 | C3×C15 | C30 | C15 | C3×Dic5 | C30 | Dic5 | C15 | C10 | C5 | C3×C6 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C15⋊C8 ►in GL6(𝔽241)
| 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 | 93 | 0 | 0 |
| 0 | 0 | 132 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 226 |
| 0 | 0 | 0 | 0 | 72 | 226 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 187 | 234 | 0 | 0 |
| 0 | 0 | 141 | 54 | 0 | 0 |
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
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