direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×Dic15, C30⋊3C12, C30⋊4Dic3, C30.54D6, C6.22D30, C62.1D5, C6⋊(C3×Dic5), (C3×C30)⋊8C4, C5⋊3(C6×Dic3), (C6×C30).2C2, (C2×C30).7S3, (C2×C30).4C6, C3⋊2(C6×Dic5), (C3×C6)⋊2Dic5, (C2×C6).6D15, C6.11(C6×D5), C2.2(C6×D15), C15⋊10(C2×C12), C22.(C3×D15), C10.11(S3×C6), C30.11(C2×C6), C10⋊2(C3×Dic3), (C3×C6).30D10, C32⋊6(C2×Dic5), C15⋊11(C2×Dic3), (C3×C30).40C22, (C2×C10).(C3×S3), (C3×C15)⋊32(C2×C4), (C2×C6).3(C3×D5), SmallGroup(360,103)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C6×Dic15 |
Generators and relations for C6×Dic15
G = < a,b,c | a6=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 204 in 74 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, C6, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C15, C3×C6, C3×C6, Dic5, C2×C10, C2×Dic3, C2×C12, C30, C30, C30, C3×Dic3, C62, C2×Dic5, C3×C15, C3×Dic5, Dic15, C2×C30, C2×C30, C6×Dic3, C3×C30, C3×C30, C6×Dic5, C2×Dic15, C3×Dic15, C6×C30, C6×Dic15
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, Dic3, C12, D6, C2×C6, C3×S3, Dic5, D10, C2×Dic3, C2×C12, C3×D5, D15, C3×Dic3, S3×C6, C2×Dic5, C3×Dic5, Dic15, C6×D5, D30, C6×Dic3, C3×D15, C6×Dic5, C2×Dic15, C3×Dic15, C6×D15, C6×Dic15
►On 120 points
Generators in S120
(1 48 21 38 11 58)(2 49 22 39 12 59)(3 50 23 40 13 60)(4 51 24 41 14 31)(5 52 25 42 15 32)(6 53 26 43 16 33)(7 54 27 44 17 34)(8 55 28 45 18 35)(9 56 29 46 19 36)(10 57 30 47 20 37)(61 101 71 111 81 91)(62 102 72 112 82 92)(63 103 73 113 83 93)(64 104 74 114 84 94)(65 105 75 115 85 95)(66 106 76 116 86 96)(67 107 77 117 87 97)(68 108 78 118 88 98)(69 109 79 119 89 99)(70 110 80 120 90 100)
(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 86 16 71)(2 85 17 70)(3 84 18 69)(4 83 19 68)(5 82 20 67)(6 81 21 66)(7 80 22 65)(8 79 23 64)(9 78 24 63)(10 77 25 62)(11 76 26 61)(12 75 27 90)(13 74 28 89)(14 73 29 88)(15 72 30 87)(31 113 46 98)(32 112 47 97)(33 111 48 96)(34 110 49 95)(35 109 50 94)(36 108 51 93)(37 107 52 92)(38 106 53 91)(39 105 54 120)(40 104 55 119)(41 103 56 118)(42 102 57 117)(43 101 58 116)(44 100 59 115)(45 99 60 114)
G:=sub<Sym(120)| (1,48,21,38,11,58)(2,49,22,39,12,59)(3,50,23,40,13,60)(4,51,24,41,14,31)(5,52,25,42,15,32)(6,53,26,43,16,33)(7,54,27,44,17,34)(8,55,28,45,18,35)(9,56,29,46,19,36)(10,57,30,47,20,37)(61,101,71,111,81,91)(62,102,72,112,82,92)(63,103,73,113,83,93)(64,104,74,114,84,94)(65,105,75,115,85,95)(66,106,76,116,86,96)(67,107,77,117,87,97)(68,108,78,118,88,98)(69,109,79,119,89,99)(70,110,80,120,90,100), (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), (1,86,16,71)(2,85,17,70)(3,84,18,69)(4,83,19,68)(5,82,20,67)(6,81,21,66)(7,80,22,65)(8,79,23,64)(9,78,24,63)(10,77,25,62)(11,76,26,61)(12,75,27,90)(13,74,28,89)(14,73,29,88)(15,72,30,87)(31,113,46,98)(32,112,47,97)(33,111,48,96)(34,110,49,95)(35,109,50,94)(36,108,51,93)(37,107,52,92)(38,106,53,91)(39,105,54,120)(40,104,55,119)(41,103,56,118)(42,102,57,117)(43,101,58,116)(44,100,59,115)(45,99,60,114)>;
G:=Group( (1,48,21,38,11,58)(2,49,22,39,12,59)(3,50,23,40,13,60)(4,51,24,41,14,31)(5,52,25,42,15,32)(6,53,26,43,16,33)(7,54,27,44,17,34)(8,55,28,45,18,35)(9,56,29,46,19,36)(10,57,30,47,20,37)(61,101,71,111,81,91)(62,102,72,112,82,92)(63,103,73,113,83,93)(64,104,74,114,84,94)(65,105,75,115,85,95)(66,106,76,116,86,96)(67,107,77,117,87,97)(68,108,78,118,88,98)(69,109,79,119,89,99)(70,110,80,120,90,100), (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), (1,86,16,71)(2,85,17,70)(3,84,18,69)(4,83,19,68)(5,82,20,67)(6,81,21,66)(7,80,22,65)(8,79,23,64)(9,78,24,63)(10,77,25,62)(11,76,26,61)(12,75,27,90)(13,74,28,89)(14,73,29,88)(15,72,30,87)(31,113,46,98)(32,112,47,97)(33,111,48,96)(34,110,49,95)(35,109,50,94)(36,108,51,93)(37,107,52,92)(38,106,53,91)(39,105,54,120)(40,104,55,119)(41,103,56,118)(42,102,57,117)(43,101,58,116)(44,100,59,115)(45,99,60,114) );
G=PermutationGroup([[(1,48,21,38,11,58),(2,49,22,39,12,59),(3,50,23,40,13,60),(4,51,24,41,14,31),(5,52,25,42,15,32),(6,53,26,43,16,33),(7,54,27,44,17,34),(8,55,28,45,18,35),(9,56,29,46,19,36),(10,57,30,47,20,37),(61,101,71,111,81,91),(62,102,72,112,82,92),(63,103,73,113,83,93),(64,104,74,114,84,94),(65,105,75,115,85,95),(66,106,76,116,86,96),(67,107,77,117,87,97),(68,108,78,118,88,98),(69,109,79,119,89,99),(70,110,80,120,90,100)], [(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)], [(1,86,16,71),(2,85,17,70),(3,84,18,69),(4,83,19,68),(5,82,20,67),(6,81,21,66),(7,80,22,65),(8,79,23,64),(9,78,24,63),(10,77,25,62),(11,76,26,61),(12,75,27,90),(13,74,28,89),(14,73,29,88),(15,72,30,87),(31,113,46,98),(32,112,47,97),(33,111,48,96),(34,110,49,95),(35,109,50,94),(36,108,51,93),(37,107,52,92),(38,106,53,91),(39,105,54,120),(40,104,55,119),(41,103,56,118),(42,102,57,117),(43,101,58,116),(44,100,59,115),(45,99,60,114)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6O | 10A | ··· | 10F | 12A | ··· | 12H | 15A | ··· | 15P | 30A | ··· | 30AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 2 | 2 | 1 | ··· | 1 | 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 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D5 | Dic3 | D6 | C3×S3 | Dic5 | D10 | C3×D5 | D15 | C3×Dic3 | S3×C6 | C3×Dic5 | Dic15 | C6×D5 | D30 | C3×D15 | C3×Dic15 | C6×D15 |
| kernel | C6×Dic15 | C3×Dic15 | C6×C30 | C2×Dic15 | C3×C30 | Dic15 | C2×C30 | C30 | C2×C30 | C62 | C30 | C30 | C2×C10 | C3×C6 | C3×C6 | C2×C6 | C2×C6 | C10 | C10 | C6 | C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 2 | 8 | 8 | 4 | 4 | 8 | 16 | 8 |
Matrix representation of C6×Dic15 ►in GL3(𝔽61) generated by
| 48 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 1 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 19 |
| 60 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 0 |
G:=sub<GL(3,GF(61))| [48,0,0,0,13,0,0,0,13],[1,0,0,0,45,0,0,0,19],[60,0,0,0,0,60,0,1,0] >;
C6×Dic15 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{15} % in TeX
G:=Group("C6xDic15"); // GroupNames label
G:=SmallGroup(360,103);
// by ID
G=gap.SmallGroup(360,103);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,1444,10373]);
// Polycyclic
G:=Group<a,b,c|a^6=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations