direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×Dic5, C30⋊4C4, C10⋊2C12, C6.16D10, C30.16C22, C5⋊3(C2×C12), (C2×C10).C6, C15⋊11(C2×C4), (C2×C6).2D5, C2.2(C6×D5), C22.(C3×D5), C10.4(C2×C6), (C2×C30).2C2, SmallGroup(120,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C6×Dic5 |
Generators and relations for C6×Dic5
G = < a,b,c | a6=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×Dic5
►Regular action on 120 points
Generators in S120
(1 49 29 39 19 59)(2 50 30 40 20 60)(3 41 21 31 11 51)(4 42 22 32 12 52)(5 43 23 33 13 53)(6 44 24 34 14 54)(7 45 25 35 15 55)(8 46 26 36 16 56)(9 47 27 37 17 57)(10 48 28 38 18 58)(61 101 81 91 71 111)(62 102 82 92 72 112)(63 103 83 93 73 113)(64 104 84 94 74 114)(65 105 85 95 75 115)(66 106 86 96 76 116)(67 107 87 97 77 117)(68 108 88 98 78 118)(69 109 89 99 79 119)(70 110 90 100 80 120)
(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 66 6 61)(2 65 7 70)(3 64 8 69)(4 63 9 68)(5 62 10 67)(11 74 16 79)(12 73 17 78)(13 72 18 77)(14 71 19 76)(15 80 20 75)(21 84 26 89)(22 83 27 88)(23 82 28 87)(24 81 29 86)(25 90 30 85)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)
G:=sub<Sym(120)| (1,49,29,39,19,59)(2,50,30,40,20,60)(3,41,21,31,11,51)(4,42,22,32,12,52)(5,43,23,33,13,53)(6,44,24,34,14,54)(7,45,25,35,15,55)(8,46,26,36,16,56)(9,47,27,37,17,57)(10,48,28,38,18,58)(61,101,81,91,71,111)(62,102,82,92,72,112)(63,103,83,93,73,113)(64,104,84,94,74,114)(65,105,85,95,75,115)(66,106,86,96,76,116)(67,107,87,97,77,117)(68,108,88,98,78,118)(69,109,89,99,79,119)(70,110,90,100,80,120), (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), (1,66,6,61)(2,65,7,70)(3,64,8,69)(4,63,9,68)(5,62,10,67)(11,74,16,79)(12,73,17,78)(13,72,18,77)(14,71,19,76)(15,80,20,75)(21,84,26,89)(22,83,27,88)(23,82,28,87)(24,81,29,86)(25,90,30,85)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,114,56,119)(52,113,57,118)(53,112,58,117)(54,111,59,116)(55,120,60,115)>;
G:=Group( (1,49,29,39,19,59)(2,50,30,40,20,60)(3,41,21,31,11,51)(4,42,22,32,12,52)(5,43,23,33,13,53)(6,44,24,34,14,54)(7,45,25,35,15,55)(8,46,26,36,16,56)(9,47,27,37,17,57)(10,48,28,38,18,58)(61,101,81,91,71,111)(62,102,82,92,72,112)(63,103,83,93,73,113)(64,104,84,94,74,114)(65,105,85,95,75,115)(66,106,86,96,76,116)(67,107,87,97,77,117)(68,108,88,98,78,118)(69,109,89,99,79,119)(70,110,90,100,80,120), (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), (1,66,6,61)(2,65,7,70)(3,64,8,69)(4,63,9,68)(5,62,10,67)(11,74,16,79)(12,73,17,78)(13,72,18,77)(14,71,19,76)(15,80,20,75)(21,84,26,89)(22,83,27,88)(23,82,28,87)(24,81,29,86)(25,90,30,85)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,114,56,119)(52,113,57,118)(53,112,58,117)(54,111,59,116)(55,120,60,115) );
G=PermutationGroup([[(1,49,29,39,19,59),(2,50,30,40,20,60),(3,41,21,31,11,51),(4,42,22,32,12,52),(5,43,23,33,13,53),(6,44,24,34,14,54),(7,45,25,35,15,55),(8,46,26,36,16,56),(9,47,27,37,17,57),(10,48,28,38,18,58),(61,101,81,91,71,111),(62,102,82,92,72,112),(63,103,83,93,73,113),(64,104,84,94,74,114),(65,105,85,95,75,115),(66,106,86,96,76,116),(67,107,87,97,77,117),(68,108,88,98,78,118),(69,109,89,99,79,119),(70,110,90,100,80,120)], [(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)], [(1,66,6,61),(2,65,7,70),(3,64,8,69),(4,63,9,68),(5,62,10,67),(11,74,16,79),(12,73,17,78),(13,72,18,77),(14,71,19,76),(15,80,20,75),(21,84,26,89),(22,83,27,88),(23,82,28,87),(24,81,29,86),(25,90,30,85),(31,94,36,99),(32,93,37,98),(33,92,38,97),(34,91,39,96),(35,100,40,95),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(51,114,56,119),(52,113,57,118),(53,112,58,117),(54,111,59,116),(55,120,60,115)]])
C6×Dic5 is a maximal subgroup of
D6⋊Dic5 D30⋊4C4 C30.Q8 Dic15⋊5C4 C6.Dic10 C15⋊8M4(2) Dic3.D10 D5×C2×C12
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6F | 10A | ··· | 10F | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D5 | Dic5 | D10 | C3×D5 | C3×Dic5 | C6×D5 |
| kernel | C6×Dic5 | C3×Dic5 | C2×C30 | C2×Dic5 | C30 | Dic5 | C2×C10 | C10 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 2 | 4 | 8 | 4 |
Matrix representation of C6×Dic5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 1 |
| 0 | 0 | 16 | 44 |
| 11 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 43 |
| 0 | 0 | 44 | 0 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,47,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,1,0,0,0,0,60,16,0,0,1,44],[11,0,0,0,0,60,0,0,0,0,0,44,0,0,43,0] >;
C6×Dic5 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_5 % in TeX
G:=Group("C6xDic5"); // GroupNames label
G:=SmallGroup(120,19);
// by ID
G=gap.SmallGroup(120,19);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-5,60,2404]);
// Polycyclic
G:=Group<a,b,c|a^6=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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