metabelian, supersoluble, monomial
Aliases: C12.59D6, C62.34C22, (C2×C12)⋊4S3, (C6×C12)⋊5C2, (C2×C6).39D6, C12⋊S3⋊8C2, C3⋊5(C4○D12), C32⋊8(C4○D4), C32⋊7D4⋊5C2, C32⋊4Q8⋊8C2, (C3×C6).32C23, C6.33(C22×S3), (C3×C12).36C22, C3⋊Dic3.17C22, (C4×C3⋊S3)⋊8C2, (C2×C4)⋊3(C3⋊S3), C4.16(C2×C3⋊S3), C22.2(C2×C3⋊S3), C2.5(C22×C3⋊S3), (C2×C3⋊S3).17C22, SmallGroup(144,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.59D6
G = < a,b,c | a12=b6=1, c2=a6, ab=ba, cac-1=a5, cbc-1=a6b-1 >
Subgroups: 378 in 120 conjugacy classes, 47 normal (15 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], S3 [×8], C6 [×4], C6 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×8], C12 [×8], D6 [×8], C2×C6 [×4], C4○D4, C3⋊S3 [×2], C3×C6, C3×C6, Dic6 [×4], C4×S3 [×8], D12 [×4], C3⋊D4 [×8], C2×C12 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C62, C4○D12 [×4], C32⋊4Q8, C4×C3⋊S3 [×2], C12⋊S3, C32⋊7D4 [×2], C6×C12, C12.59D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], C4○D12 [×4], C22×C3⋊S3, C12.59D6
►On 72 points
(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)
(1 66 20 38 54 34)(2 67 21 39 55 35)(3 68 22 40 56 36)(4 69 23 41 57 25)(5 70 24 42 58 26)(6 71 13 43 59 27)(7 72 14 44 60 28)(8 61 15 45 49 29)(9 62 16 46 50 30)(10 63 17 47 51 31)(11 64 18 48 52 32)(12 65 19 37 53 33)
(1 23 7 17)(2 16 8 22)(3 21 9 15)(4 14 10 20)(5 19 11 13)(6 24 12 18)(25 38 31 44)(26 43 32 37)(27 48 33 42)(28 41 34 47)(29 46 35 40)(30 39 36 45)(49 56 55 50)(51 54 57 60)(52 59 58 53)(61 62 67 68)(63 72 69 66)(64 65 70 71)
G:=sub<Sym(72)| (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), (1,66,20,38,54,34)(2,67,21,39,55,35)(3,68,22,40,56,36)(4,69,23,41,57,25)(5,70,24,42,58,26)(6,71,13,43,59,27)(7,72,14,44,60,28)(8,61,15,45,49,29)(9,62,16,46,50,30)(10,63,17,47,51,31)(11,64,18,48,52,32)(12,65,19,37,53,33), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,38,31,44)(26,43,32,37)(27,48,33,42)(28,41,34,47)(29,46,35,40)(30,39,36,45)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,62,67,68)(63,72,69,66)(64,65,70,71)>;
G:=Group( (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), (1,66,20,38,54,34)(2,67,21,39,55,35)(3,68,22,40,56,36)(4,69,23,41,57,25)(5,70,24,42,58,26)(6,71,13,43,59,27)(7,72,14,44,60,28)(8,61,15,45,49,29)(9,62,16,46,50,30)(10,63,17,47,51,31)(11,64,18,48,52,32)(12,65,19,37,53,33), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,38,31,44)(26,43,32,37)(27,48,33,42)(28,41,34,47)(29,46,35,40)(30,39,36,45)(49,56,55,50)(51,54,57,60)(52,59,58,53)(61,62,67,68)(63,72,69,66)(64,65,70,71) );
G=PermutationGroup([(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)], [(1,66,20,38,54,34),(2,67,21,39,55,35),(3,68,22,40,56,36),(4,69,23,41,57,25),(5,70,24,42,58,26),(6,71,13,43,59,27),(7,72,14,44,60,28),(8,61,15,45,49,29),(9,62,16,46,50,30),(10,63,17,47,51,31),(11,64,18,48,52,32),(12,65,19,37,53,33)], [(1,23,7,17),(2,16,8,22),(3,21,9,15),(4,14,10,20),(5,19,11,13),(6,24,12,18),(25,38,31,44),(26,43,32,37),(27,48,33,42),(28,41,34,47),(29,46,35,40),(30,39,36,45),(49,56,55,50),(51,54,57,60),(52,59,58,53),(61,62,67,68),(63,72,69,66),(64,65,70,71)])
C12.59D6 is a maximal subgroup of
C12.80D12 C122⋊C2 C62.37D4 C3⋊C8.22D6 D12.27D6 D12.28D6 Dic6.29D6 C24.95D6 C24.78D6 C24.47D6 C24⋊3D6 C24.5D6 C62.131D4 C62.134D4 C62.74D4 D12.33D6 S3×C4○D12 D12⋊24D6 C32⋊82+ 1+4 C32⋊72- 1+4 C4○D4×C3⋊S3 C62.154C23 C32⋊92- 1+4 C62.36D6 C36.70D6 C12.73S32 C12.57S32 C12.58S32 C62.93D6 C62.160D6
C12.59D6 is a maximal quotient of
C4×C32⋊4Q8 C12.25Dic6 C122⋊16C2 C4×C12⋊S3 C122⋊6C2 C122⋊2C2 C62.223C23 C62.227C23 C62.228C23 C62.229C23 C62.233C23 C62.238C23 C62.240C23 C62.242C23 C62⋊10Q8 C62.247C23 C4×C32⋊7D4 C62.129D4 C62⋊19D4 C36.70D6 C62.47D6 C12.73S32 C12.57S32 C12.58S32 C62.93D6 C62.160D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 |
| kernel | C12.59D6 | C32⋊4Q8 | C4×C3⋊S3 | C12⋊S3 | C32⋊7D4 | C6×C12 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 4 | 2 | 16 |
Matrix representation of C12.59D6 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,4,2,0,0,11,2],[1,12,0,0,0,12,0,0,0,0,5,0,0,0,8,8] >;
C12.59D6 in GAP, Magma, Sage, TeX
C_{12}._{59}D_6 % in TeX
G:=Group("C12.59D6"); // GroupNames label
G:=SmallGroup(144,171);
// by ID
G=gap.SmallGroup(144,171);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^12=b^6=1,c^2=a^6,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^6*b^-1>;
// generators/relations