direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C15⋊3C8, C15⋊3C24, C60.2C6, C30.5C12, C60.11S3, C12.8D15, C30.8Dic3, C6.4Dic15, C15⋊4(C3⋊C8), (C3×C15)⋊11C8, C6.(C3×Dic5), C20.2(C3×S3), (C3×C30).8C4, (C3×C60).4C2, C12.2(C3×D5), (C3×C12).3D5, C4.2(C3×D15), C2.(C3×Dic15), C32⋊3(C5⋊2C8), (C3×C6).2Dic5, C10.2(C3×Dic3), C5⋊2(C3×C3⋊C8), C3⋊(C3×C5⋊2C8), SmallGroup(360,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C3×C15⋊3C8 |
Generators and relations for C3×C15⋊3C8
G = < a,b,c | a3=b15=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C15⋊3C8
►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)
(1 110 54 80 27 91 40 62)(2 109 55 79 28 105 41 61)(3 108 56 78 29 104 42 75)(4 107 57 77 30 103 43 74)(5 106 58 76 16 102 44 73)(6 120 59 90 17 101 45 72)(7 119 60 89 18 100 31 71)(8 118 46 88 19 99 32 70)(9 117 47 87 20 98 33 69)(10 116 48 86 21 97 34 68)(11 115 49 85 22 96 35 67)(12 114 50 84 23 95 36 66)(13 113 51 83 24 94 37 65)(14 112 52 82 25 93 38 64)(15 111 53 81 26 92 39 63)
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), (1,110,54,80,27,91,40,62)(2,109,55,79,28,105,41,61)(3,108,56,78,29,104,42,75)(4,107,57,77,30,103,43,74)(5,106,58,76,16,102,44,73)(6,120,59,90,17,101,45,72)(7,119,60,89,18,100,31,71)(8,118,46,88,19,99,32,70)(9,117,47,87,20,98,33,69)(10,116,48,86,21,97,34,68)(11,115,49,85,22,96,35,67)(12,114,50,84,23,95,36,66)(13,113,51,83,24,94,37,65)(14,112,52,82,25,93,38,64)(15,111,53,81,26,92,39,63)>;
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), (1,110,54,80,27,91,40,62)(2,109,55,79,28,105,41,61)(3,108,56,78,29,104,42,75)(4,107,57,77,30,103,43,74)(5,106,58,76,16,102,44,73)(6,120,59,90,17,101,45,72)(7,119,60,89,18,100,31,71)(8,118,46,88,19,99,32,70)(9,117,47,87,20,98,33,69)(10,116,48,86,21,97,34,68)(11,115,49,85,22,96,35,67)(12,114,50,84,23,95,36,66)(13,113,51,83,24,94,37,65)(14,112,52,82,25,93,38,64)(15,111,53,81,26,92,39,63) );
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)], [(1,110,54,80,27,91,40,62),(2,109,55,79,28,105,41,61),(3,108,56,78,29,104,42,75),(4,107,57,77,30,103,43,74),(5,106,58,76,16,102,44,73),(6,120,59,90,17,101,45,72),(7,119,60,89,18,100,31,71),(8,118,46,88,19,99,32,70),(9,117,47,87,20,98,33,69),(10,116,48,86,21,97,34,68),(11,115,49,85,22,96,35,67),(12,114,50,84,23,95,36,66),(13,113,51,83,24,94,37,65),(14,112,52,82,25,93,38,64),(15,111,53,81,26,92,39,63)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | ··· | 15P | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | ··· | 30P | 60A | ··· | 60AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | ||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | D5 | Dic3 | C3×S3 | Dic5 | C3⋊C8 | C3×D5 | D15 | C3×Dic3 | C5⋊2C8 | C3×Dic5 | Dic15 | C3×C3⋊C8 | C3×D15 | C3×C5⋊2C8 | C15⋊3C8 | C3×Dic15 | C3×C15⋊3C8 |
| kernel | C3×C15⋊3C8 | C3×C60 | C15⋊3C8 | C3×C30 | C60 | C3×C15 | C30 | C15 | C60 | C3×C12 | C30 | C20 | C3×C6 | C15 | C12 | C12 | C10 | C32 | C6 | C6 | C5 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 |
Matrix representation of C3×C15⋊3C8 ►in GL5(𝔽241)
| 15 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 225 | 0 | 0 |
| 0 | 0 | 0 | 178 | 190 |
| 0 | 0 | 0 | 179 | 11 |
| 233 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 224 | 91 |
| 0 | 0 | 0 | 18 | 17 |
G:=sub<GL(5,GF(241))| [15,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,15,0,0,0,0,0,225,0,0,0,0,0,178,179,0,0,0,190,11],[233,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,224,18,0,0,0,91,17] >;
C3×C15⋊3C8 in GAP, Magma, Sage, TeX
C_3\times C_{15}\rtimes_3C_8 % in TeX
G:=Group("C3xC15:3C8"); // GroupNames label
G:=SmallGroup(360,35);
// by ID
G=gap.SmallGroup(360,35);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,-5,36,50,1444,10373]);
// 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^-1>;
// generators/relations
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