direct product, abelian, monomial
Aliases: C6×C18, SmallGroup(108,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C18 |
| C1 — C6×C18 |
| C1 — C6×C18 |
Generators and relations for C6×C18
G = < a,b | a6=b18=1, ab=ba >
Smallest permutation representation of C6×C18
►Regular action on 108 points
Generators in S108
(1 45 56 108 89 31)(2 46 57 91 90 32)(3 47 58 92 73 33)(4 48 59 93 74 34)(5 49 60 94 75 35)(6 50 61 95 76 36)(7 51 62 96 77 19)(8 52 63 97 78 20)(9 53 64 98 79 21)(10 54 65 99 80 22)(11 37 66 100 81 23)(12 38 67 101 82 24)(13 39 68 102 83 25)(14 40 69 103 84 26)(15 41 70 104 85 27)(16 42 71 105 86 28)(17 43 72 106 87 29)(18 44 55 107 88 30)
(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)
G:=sub<Sym(108)| (1,45,56,108,89,31)(2,46,57,91,90,32)(3,47,58,92,73,33)(4,48,59,93,74,34)(5,49,60,94,75,35)(6,50,61,95,76,36)(7,51,62,96,77,19)(8,52,63,97,78,20)(9,53,64,98,79,21)(10,54,65,99,80,22)(11,37,66,100,81,23)(12,38,67,101,82,24)(13,39,68,102,83,25)(14,40,69,103,84,26)(15,41,70,104,85,27)(16,42,71,105,86,28)(17,43,72,106,87,29)(18,44,55,107,88,30), (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)>;
G:=Group( (1,45,56,108,89,31)(2,46,57,91,90,32)(3,47,58,92,73,33)(4,48,59,93,74,34)(5,49,60,94,75,35)(6,50,61,95,76,36)(7,51,62,96,77,19)(8,52,63,97,78,20)(9,53,64,98,79,21)(10,54,65,99,80,22)(11,37,66,100,81,23)(12,38,67,101,82,24)(13,39,68,102,83,25)(14,40,69,103,84,26)(15,41,70,104,85,27)(16,42,71,105,86,28)(17,43,72,106,87,29)(18,44,55,107,88,30), (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) );
G=PermutationGroup([[(1,45,56,108,89,31),(2,46,57,91,90,32),(3,47,58,92,73,33),(4,48,59,93,74,34),(5,49,60,94,75,35),(6,50,61,95,76,36),(7,51,62,96,77,19),(8,52,63,97,78,20),(9,53,64,98,79,21),(10,54,65,99,80,22),(11,37,66,100,81,23),(12,38,67,101,82,24),(13,39,68,102,83,25),(14,40,69,103,84,26),(15,41,70,104,85,27),(16,42,71,105,86,28),(17,43,72,106,87,29),(18,44,55,107,88,30)], [(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)]])
C6×C18 is a maximal subgroup of
C6.D18 C62.C9 C62.11C32 C62.12C32 C62.13C32 C62.14C32 C62.15C32 C62.16C32 C62.25C32
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6X | 9A | ··· | 9R | 18A | ··· | 18BB |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 |
| kernel | C6×C18 | C3×C18 | C2×C18 | C62 | C18 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 6 | 2 | 18 | 6 | 18 | 54 |
Matrix representation of C6×C18 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 1 |
| 14 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(19))| [18,0,0,0,11,0,0,0,1],[14,0,0,0,5,0,0,0,3] >;
C6×C18 in GAP, Magma, Sage, TeX
C_6\times C_{18} % in TeX
G:=Group("C6xC18"); // GroupNames label
G:=SmallGroup(108,29);
// by ID
G=gap.SmallGroup(108,29);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,147]);
// Polycyclic
G:=Group<a,b|a^6=b^18=1,a*b=b*a>;
// generators/relations
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