direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic6, C15⋊4Q8, C60.4C2, C20.3S3, C12.1C10, C10.13D6, Dic3.C10, C30.18C22, C3⋊(C5×Q8), C4.(C5×S3), C2.3(S3×C10), C6.1(C2×C10), (C5×Dic3).2C2, SmallGroup(120,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic6
G = < a,b,c | a5=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic6
►Regular action on 120 points
Generators in S120
(1 16 98 75 118)(2 17 99 76 119)(3 18 100 77 120)(4 19 101 78 109)(5 20 102 79 110)(6 21 103 80 111)(7 22 104 81 112)(8 23 105 82 113)(9 24 106 83 114)(10 13 107 84 115)(11 14 108 73 116)(12 15 97 74 117)(25 52 95 46 72)(26 53 96 47 61)(27 54 85 48 62)(28 55 86 37 63)(29 56 87 38 64)(30 57 88 39 65)(31 58 89 40 66)(32 59 90 41 67)(33 60 91 42 68)(34 49 92 43 69)(35 50 93 44 70)(36 51 94 45 71)
(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 38 7 44)(2 37 8 43)(3 48 9 42)(4 47 10 41)(5 46 11 40)(6 45 12 39)(13 67 19 61)(14 66 20 72)(15 65 21 71)(16 64 22 70)(17 63 23 69)(18 62 24 68)(25 108 31 102)(26 107 32 101)(27 106 33 100)(28 105 34 99)(29 104 35 98)(30 103 36 97)(49 76 55 82)(50 75 56 81)(51 74 57 80)(52 73 58 79)(53 84 59 78)(54 83 60 77)(85 114 91 120)(86 113 92 119)(87 112 93 118)(88 111 94 117)(89 110 95 116)(90 109 96 115)
G:=sub<Sym(120)| (1,16,98,75,118)(2,17,99,76,119)(3,18,100,77,120)(4,19,101,78,109)(5,20,102,79,110)(6,21,103,80,111)(7,22,104,81,112)(8,23,105,82,113)(9,24,106,83,114)(10,13,107,84,115)(11,14,108,73,116)(12,15,97,74,117)(25,52,95,46,72)(26,53,96,47,61)(27,54,85,48,62)(28,55,86,37,63)(29,56,87,38,64)(30,57,88,39,65)(31,58,89,40,66)(32,59,90,41,67)(33,60,91,42,68)(34,49,92,43,69)(35,50,93,44,70)(36,51,94,45,71), (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), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,67,19,61)(14,66,20,72)(15,65,21,71)(16,64,22,70)(17,63,23,69)(18,62,24,68)(25,108,31,102)(26,107,32,101)(27,106,33,100)(28,105,34,99)(29,104,35,98)(30,103,36,97)(49,76,55,82)(50,75,56,81)(51,74,57,80)(52,73,58,79)(53,84,59,78)(54,83,60,77)(85,114,91,120)(86,113,92,119)(87,112,93,118)(88,111,94,117)(89,110,95,116)(90,109,96,115)>;
G:=Group( (1,16,98,75,118)(2,17,99,76,119)(3,18,100,77,120)(4,19,101,78,109)(5,20,102,79,110)(6,21,103,80,111)(7,22,104,81,112)(8,23,105,82,113)(9,24,106,83,114)(10,13,107,84,115)(11,14,108,73,116)(12,15,97,74,117)(25,52,95,46,72)(26,53,96,47,61)(27,54,85,48,62)(28,55,86,37,63)(29,56,87,38,64)(30,57,88,39,65)(31,58,89,40,66)(32,59,90,41,67)(33,60,91,42,68)(34,49,92,43,69)(35,50,93,44,70)(36,51,94,45,71), (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), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,67,19,61)(14,66,20,72)(15,65,21,71)(16,64,22,70)(17,63,23,69)(18,62,24,68)(25,108,31,102)(26,107,32,101)(27,106,33,100)(28,105,34,99)(29,104,35,98)(30,103,36,97)(49,76,55,82)(50,75,56,81)(51,74,57,80)(52,73,58,79)(53,84,59,78)(54,83,60,77)(85,114,91,120)(86,113,92,119)(87,112,93,118)(88,111,94,117)(89,110,95,116)(90,109,96,115) );
G=PermutationGroup([[(1,16,98,75,118),(2,17,99,76,119),(3,18,100,77,120),(4,19,101,78,109),(5,20,102,79,110),(6,21,103,80,111),(7,22,104,81,112),(8,23,105,82,113),(9,24,106,83,114),(10,13,107,84,115),(11,14,108,73,116),(12,15,97,74,117),(25,52,95,46,72),(26,53,96,47,61),(27,54,85,48,62),(28,55,86,37,63),(29,56,87,38,64),(30,57,88,39,65),(31,58,89,40,66),(32,59,90,41,67),(33,60,91,42,68),(34,49,92,43,69),(35,50,93,44,70),(36,51,94,45,71)], [(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)], [(1,38,7,44),(2,37,8,43),(3,48,9,42),(4,47,10,41),(5,46,11,40),(6,45,12,39),(13,67,19,61),(14,66,20,72),(15,65,21,71),(16,64,22,70),(17,63,23,69),(18,62,24,68),(25,108,31,102),(26,107,32,101),(27,106,33,100),(28,105,34,99),(29,104,35,98),(30,103,36,97),(49,76,55,82),(50,75,56,81),(51,74,57,80),(52,73,58,79),(53,84,59,78),(54,83,60,77),(85,114,91,120),(86,113,92,119),(87,112,93,118),(88,111,94,117),(89,110,95,116),(90,109,96,115)]])
C5×Dic6 is a maximal subgroup of
C30.D4 Dic6⋊D5 C15⋊Q16 C5⋊Dic12 D20⋊S3 D15⋊Q8 C12.28D10 C5×S3×Q8
45 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | Q8 | D6 | Dic6 | C5×S3 | C5×Q8 | S3×C10 | C5×Dic6 |
| kernel | C5×Dic6 | C5×Dic3 | C60 | Dic6 | Dic3 | C12 | C20 | C15 | C10 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C5×Dic6 ►in GL2(𝔽11) generated by
| 9 | 0 |
| 0 | 9 |
| 1 | 2 |
| 7 | 4 |
| 10 | 10 |
| 2 | 1 |
G:=sub<GL(2,GF(11))| [9,0,0,9],[1,7,2,4],[10,2,10,1] >;
C5×Dic6 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_6 % in TeX
G:=Group("C5xDic6"); // GroupNames label
G:=SmallGroup(120,21);
// by ID
G=gap.SmallGroup(120,21);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-3,100,221,106,2004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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