direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5xDic6, C15:4Q8, C60.4C2, C20.3S3, C12.1C10, C10.13D6, Dic3.C10, C30.18C22, C3:(C5xQ8), C4.(C5xS3), C2.3(S3xC10), C6.1(C2xC10), (C5xDic3).2C2, SmallGroup(120,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5xDic6
G = < a,b,c | a5=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Quotients: C1, C2, C22, C5, S3, Q8, C10, D6, C2xC10, Dic6, C5xS3, C5xQ8, S3xC10, C5xDic6
Smallest permutation representation of C5xDic6
►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)]])
C5xDic6 is a maximal subgroup of
C30.D4 Dic6:D5 C15:Q16 C5:Dic12 D20:S3 D15:Q8 C12.28D10 C5xS3xQ8
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 | C5xS3 | C5xQ8 | S3xC10 | C5xDic6 |
| kernel | C5xDic6 | C5xDic3 | 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 C5xDic6 ►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] >;
C5xDic6 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
Export