metabelian, supersoluble, monomial
Aliases: C6.18D18, C18.18D6, C62.12S3, (C3×C9)⋊9D4, (C2×C6)⋊4D9, (C6×C18)⋊5C2, (C2×C18)⋊4S3, C9⋊3(C3⋊D4), C3⋊3(C9⋊D4), C9⋊Dic3⋊4C2, (C3×C6).54D6, C22⋊3(C9⋊S3), C3.(C32⋊7D4), (C3×C18).22C22, C32.5(C3⋊D4), (C2×C9⋊S3)⋊4C2, C2.5(C2×C9⋊S3), C6.12(C2×C3⋊S3), (C2×C6).6(C3⋊S3), SmallGroup(216,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D18
G = < a,b,c | a6=b18=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b-1 >
Subgroups: 430 in 80 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, D4, C9, C32, Dic3, D6, C2×C6, C2×C6, D9, C18, C18, C3⋊S3, C3×C6, C3×C6, C3⋊D4, C3×C9, Dic9, D18, C2×C18, C3⋊Dic3, C2×C3⋊S3, C62, C9⋊S3, C3×C18, C3×C18, C9⋊D4, C32⋊7D4, C9⋊Dic3, C2×C9⋊S3, C6×C18, C6.D18
Quotients: C1, C2, C22, S3, D4, D6, D9, C3⋊S3, C3⋊D4, D18, C2×C3⋊S3, C9⋊S3, C9⋊D4, C32⋊7D4, C2×C9⋊S3, C6.D18
►On 108 points
(1 65 104 48 78 22)(2 66 105 49 79 23)(3 67 106 50 80 24)(4 68 107 51 81 25)(5 69 108 52 82 26)(6 70 91 53 83 27)(7 71 92 54 84 28)(8 72 93 37 85 29)(9 55 94 38 86 30)(10 56 95 39 87 31)(11 57 96 40 88 32)(12 58 97 41 89 33)(13 59 98 42 90 34)(14 60 99 43 73 35)(15 61 100 44 74 36)(16 62 101 45 75 19)(17 63 102 46 76 20)(18 64 103 47 77 21)
(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)
(1 18 48 47)(2 46 49 17)(3 16 50 45)(4 44 51 15)(5 14 52 43)(6 42 53 13)(7 12 54 41)(8 40 37 11)(9 10 38 39)(19 80 101 67)(20 66 102 79)(21 78 103 65)(22 64 104 77)(23 76 105 63)(24 62 106 75)(25 74 107 61)(26 60 108 73)(27 90 91 59)(28 58 92 89)(29 88 93 57)(30 56 94 87)(31 86 95 55)(32 72 96 85)(33 84 97 71)(34 70 98 83)(35 82 99 69)(36 68 100 81)
G:=sub<Sym(108)| (1,65,104,48,78,22)(2,66,105,49,79,23)(3,67,106,50,80,24)(4,68,107,51,81,25)(5,69,108,52,82,26)(6,70,91,53,83,27)(7,71,92,54,84,28)(8,72,93,37,85,29)(9,55,94,38,86,30)(10,56,95,39,87,31)(11,57,96,40,88,32)(12,58,97,41,89,33)(13,59,98,42,90,34)(14,60,99,43,73,35)(15,61,100,44,74,36)(16,62,101,45,75,19)(17,63,102,46,76,20)(18,64,103,47,77,21), (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), (1,18,48,47)(2,46,49,17)(3,16,50,45)(4,44,51,15)(5,14,52,43)(6,42,53,13)(7,12,54,41)(8,40,37,11)(9,10,38,39)(19,80,101,67)(20,66,102,79)(21,78,103,65)(22,64,104,77)(23,76,105,63)(24,62,106,75)(25,74,107,61)(26,60,108,73)(27,90,91,59)(28,58,92,89)(29,88,93,57)(30,56,94,87)(31,86,95,55)(32,72,96,85)(33,84,97,71)(34,70,98,83)(35,82,99,69)(36,68,100,81)>;
G:=Group( (1,65,104,48,78,22)(2,66,105,49,79,23)(3,67,106,50,80,24)(4,68,107,51,81,25)(5,69,108,52,82,26)(6,70,91,53,83,27)(7,71,92,54,84,28)(8,72,93,37,85,29)(9,55,94,38,86,30)(10,56,95,39,87,31)(11,57,96,40,88,32)(12,58,97,41,89,33)(13,59,98,42,90,34)(14,60,99,43,73,35)(15,61,100,44,74,36)(16,62,101,45,75,19)(17,63,102,46,76,20)(18,64,103,47,77,21), (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), (1,18,48,47)(2,46,49,17)(3,16,50,45)(4,44,51,15)(5,14,52,43)(6,42,53,13)(7,12,54,41)(8,40,37,11)(9,10,38,39)(19,80,101,67)(20,66,102,79)(21,78,103,65)(22,64,104,77)(23,76,105,63)(24,62,106,75)(25,74,107,61)(26,60,108,73)(27,90,91,59)(28,58,92,89)(29,88,93,57)(30,56,94,87)(31,86,95,55)(32,72,96,85)(33,84,97,71)(34,70,98,83)(35,82,99,69)(36,68,100,81) );
G=PermutationGroup([[(1,65,104,48,78,22),(2,66,105,49,79,23),(3,67,106,50,80,24),(4,68,107,51,81,25),(5,69,108,52,82,26),(6,70,91,53,83,27),(7,71,92,54,84,28),(8,72,93,37,85,29),(9,55,94,38,86,30),(10,56,95,39,87,31),(11,57,96,40,88,32),(12,58,97,41,89,33),(13,59,98,42,90,34),(14,60,99,43,73,35),(15,61,100,44,74,36),(16,62,101,45,75,19),(17,63,102,46,76,20),(18,64,103,47,77,21)], [(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)], [(1,18,48,47),(2,46,49,17),(3,16,50,45),(4,44,51,15),(5,14,52,43),(6,42,53,13),(7,12,54,41),(8,40,37,11),(9,10,38,39),(19,80,101,67),(20,66,102,79),(21,78,103,65),(22,64,104,77),(23,76,105,63),(24,62,106,75),(25,74,107,61),(26,60,108,73),(27,90,91,59),(28,58,92,89),(29,88,93,57),(30,56,94,87),(31,86,95,55),(32,72,96,85),(33,84,97,71),(34,70,98,83),(35,82,99,69),(36,68,100,81)]])
C6.D18 is a maximal subgroup of
D18.3D6 Dic3.D18 S3×C9⋊D4 D9×C3⋊D4 C36.70D6 D4×C9⋊S3 C36.27D6
C6.D18 is a maximal quotient of C6.Dic18 C6.11D36 C36.17D6 C36.18D6 C36.19D6 C36.20D6 C62.127D6
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | ··· | 6L | 9A | ··· | 9I | 18A | ··· | 18AA |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 2 | 54 | 2 | 2 | 2 | 2 | 54 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D9 | C3⋊D4 | C3⋊D4 | D18 | C9⋊D4 |
| kernel | C6.D18 | C9⋊Dic3 | C2×C9⋊S3 | C6×C18 | C2×C18 | C62 | C3×C9 | C18 | C3×C6 | C2×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 1 | 9 | 6 | 2 | 9 | 18 |
Matrix representation of C6.D18 ►in GL4(𝔽37) generated by
| 36 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 24 | 26 |
| 36 | 36 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 11 |
| 0 | 0 | 35 | 24 |
G:=sub<GL(4,GF(37))| [36,1,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[36,1,0,0,36,0,0,0,0,0,2,24,0,0,13,26],[36,0,0,0,36,1,0,0,0,0,13,35,0,0,11,24] >;
C6.D18 in GAP, Magma, Sage, TeX
C_6.D_{18} % in TeX
G:=Group("C6.D18"); // GroupNames label
G:=SmallGroup(216,70);
// by ID
G=gap.SmallGroup(216,70);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,2115,453,1444,5189]);
// Polycyclic
G:=Group<a,b,c|a^6=b^18=1,c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations