direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×Dic3, C6⋊C20, C30⋊5C4, C10.16D6, C30.21C22, C3⋊2(C2×C20), (C2×C6).C10, C15⋊12(C2×C4), C22.(C5×S3), C2.2(S3×C10), (C2×C10).2S3, (C2×C30).3C2, C6.4(C2×C10), SmallGroup(120,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C10×Dic3 |
Generators and relations for C10×Dic3
G = < a,b,c | a10=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C10×Dic3
►Regular action 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 75 85 105 95 70)(2 76 86 106 96 61)(3 77 87 107 97 62)(4 78 88 108 98 63)(5 79 89 109 99 64)(6 80 90 110 100 65)(7 71 81 101 91 66)(8 72 82 102 92 67)(9 73 83 103 93 68)(10 74 84 104 94 69)(11 51 41 115 40 29)(12 52 42 116 31 30)(13 53 43 117 32 21)(14 54 44 118 33 22)(15 55 45 119 34 23)(16 56 46 120 35 24)(17 57 47 111 36 25)(18 58 48 112 37 26)(19 59 49 113 38 27)(20 60 50 114 39 28)
(1 16 105 120)(2 17 106 111)(3 18 107 112)(4 19 108 113)(5 20 109 114)(6 11 110 115)(7 12 101 116)(8 13 102 117)(9 14 103 118)(10 15 104 119)(21 92 43 72)(22 93 44 73)(23 94 45 74)(24 95 46 75)(25 96 47 76)(26 97 48 77)(27 98 49 78)(28 99 50 79)(29 100 41 80)(30 91 42 71)(31 66 52 81)(32 67 53 82)(33 68 54 83)(34 69 55 84)(35 70 56 85)(36 61 57 86)(37 62 58 87)(38 63 59 88)(39 64 60 89)(40 65 51 90)
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), (1,75,85,105,95,70)(2,76,86,106,96,61)(3,77,87,107,97,62)(4,78,88,108,98,63)(5,79,89,109,99,64)(6,80,90,110,100,65)(7,71,81,101,91,66)(8,72,82,102,92,67)(9,73,83,103,93,68)(10,74,84,104,94,69)(11,51,41,115,40,29)(12,52,42,116,31,30)(13,53,43,117,32,21)(14,54,44,118,33,22)(15,55,45,119,34,23)(16,56,46,120,35,24)(17,57,47,111,36,25)(18,58,48,112,37,26)(19,59,49,113,38,27)(20,60,50,114,39,28), (1,16,105,120)(2,17,106,111)(3,18,107,112)(4,19,108,113)(5,20,109,114)(6,11,110,115)(7,12,101,116)(8,13,102,117)(9,14,103,118)(10,15,104,119)(21,92,43,72)(22,93,44,73)(23,94,45,74)(24,95,46,75)(25,96,47,76)(26,97,48,77)(27,98,49,78)(28,99,50,79)(29,100,41,80)(30,91,42,71)(31,66,52,81)(32,67,53,82)(33,68,54,83)(34,69,55,84)(35,70,56,85)(36,61,57,86)(37,62,58,87)(38,63,59,88)(39,64,60,89)(40,65,51,90)>;
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), (1,75,85,105,95,70)(2,76,86,106,96,61)(3,77,87,107,97,62)(4,78,88,108,98,63)(5,79,89,109,99,64)(6,80,90,110,100,65)(7,71,81,101,91,66)(8,72,82,102,92,67)(9,73,83,103,93,68)(10,74,84,104,94,69)(11,51,41,115,40,29)(12,52,42,116,31,30)(13,53,43,117,32,21)(14,54,44,118,33,22)(15,55,45,119,34,23)(16,56,46,120,35,24)(17,57,47,111,36,25)(18,58,48,112,37,26)(19,59,49,113,38,27)(20,60,50,114,39,28), (1,16,105,120)(2,17,106,111)(3,18,107,112)(4,19,108,113)(5,20,109,114)(6,11,110,115)(7,12,101,116)(8,13,102,117)(9,14,103,118)(10,15,104,119)(21,92,43,72)(22,93,44,73)(23,94,45,74)(24,95,46,75)(25,96,47,76)(26,97,48,77)(27,98,49,78)(28,99,50,79)(29,100,41,80)(30,91,42,71)(31,66,52,81)(32,67,53,82)(33,68,54,83)(34,69,55,84)(35,70,56,85)(36,61,57,86)(37,62,58,87)(38,63,59,88)(39,64,60,89)(40,65,51,90) );
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)], [(1,75,85,105,95,70),(2,76,86,106,96,61),(3,77,87,107,97,62),(4,78,88,108,98,63),(5,79,89,109,99,64),(6,80,90,110,100,65),(7,71,81,101,91,66),(8,72,82,102,92,67),(9,73,83,103,93,68),(10,74,84,104,94,69),(11,51,41,115,40,29),(12,52,42,116,31,30),(13,53,43,117,32,21),(14,54,44,118,33,22),(15,55,45,119,34,23),(16,56,46,120,35,24),(17,57,47,111,36,25),(18,58,48,112,37,26),(19,59,49,113,38,27),(20,60,50,114,39,28)], [(1,16,105,120),(2,17,106,111),(3,18,107,112),(4,19,108,113),(5,20,109,114),(6,11,110,115),(7,12,101,116),(8,13,102,117),(9,14,103,118),(10,15,104,119),(21,92,43,72),(22,93,44,73),(23,94,45,74),(24,95,46,75),(25,96,47,76),(26,97,48,77),(27,98,49,78),(28,99,50,79),(29,100,41,80),(30,91,42,71),(31,66,52,81),(32,67,53,82),(33,68,54,83),(34,69,55,84),(35,70,56,85),(36,61,57,86),(37,62,58,87),(38,63,59,88),(39,64,60,89),(40,65,51,90)]])
C10×Dic3 is a maximal subgroup of
D10⋊Dic3 D30⋊4C4 C30.Q8 Dic15⋊5C4 C6.Dic10 Dic5.D6 S3×C2×C20
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | Dic3 | D6 | C5×S3 | C5×Dic3 | S3×C10 |
| kernel | C10×Dic3 | C5×Dic3 | C2×C30 | C30 | C2×Dic3 | Dic3 | C2×C6 | C6 | C2×C10 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 2 | 1 | 4 | 8 | 4 |
Matrix representation of C10×Dic3 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 1 |
| 0 | 0 | 60 | 0 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 1 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,20,0,0,0,0,20],[60,0,0,0,0,60,0,0,0,0,60,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,60,60,0,0,0,1] >;
C10×Dic3 in GAP, Magma, Sage, TeX
C_{10}\times {\rm Dic}_3 % in TeX
G:=Group("C10xDic3"); // GroupNames label
G:=SmallGroup(120,24);
// by ID
G=gap.SmallGroup(120,24);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-3,100,2004]);
// Polycyclic
G:=Group<a,b,c|a^10=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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