direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C12.29D6, C3⋊S3⋊3C24, C3⋊1(S3×C24), C33⋊9(C2×C8), C12.102S32, C6.1(S3×C12), C12.45(S3×C6), C32⋊6(C2×C24), C32⋊11(S3×C8), (C3×C12).179D6, C3⋊Dic3.4C12, C6.23(C6.D6), (C32×C12).61C22, (C3×C3⋊C8)⋊6C6, C3⋊C8⋊6(C3×S3), (C3×C3⋊S3)⋊3C8, (C3×C3⋊C8)⋊13S3, C4.14(C3×S32), (C6×C3⋊S3).3C4, (C4×C3⋊S3).6C6, (C32×C3⋊C8)⋊9C2, (C12×C3⋊S3).9C2, (C2×C3⋊S3).4C12, (C3×C6).58(C4×S3), (C3×C12).62(C2×C6), (C3×C6).18(C2×C12), (C3×C3⋊Dic3).3C4, C2.1(C3×C6.D6), (C32×C6).23(C2×C4), SmallGroup(432,415)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C12.29D6 |
Generators and relations for C3×C12.29D6
G = < a,b,c,d | a3=b12=1, c6=b3, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c5 >
Subgroups: 384 in 126 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C12.29D6, S3×C24, C32×C3⋊C8, C12×C3⋊S3, C3×C12.29D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C24, C4×S3, C2×C12, S32, S3×C6, S3×C8, C2×C24, C6.D6, S3×C12, C3×S32, C12.29D6, S3×C24, C3×C6.D6, C3×C12.29D6
►On 48 points
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 3 5 7 9 11 13 15 17 19 21 23)(2 12 22 8 18 4 14 24 10 20 6 16)(25 35 45 31 41 27 37 47 33 43 29 39)(26 28 30 32 34 36 38 40 42 44 46 48)
(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)
(1 26 13 38)(2 43 14 31)(3 36 15 48)(4 29 16 41)(5 46 17 34)(6 39 18 27)(7 32 19 44)(8 25 20 37)(9 42 21 30)(10 35 22 47)(11 28 23 40)(12 45 24 33)
G:=sub<Sym(48)| (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,3,5,7,9,11,13,15,17,19,21,23)(2,12,22,8,18,4,14,24,10,20,6,16)(25,35,45,31,41,27,37,47,33,43,29,39)(26,28,30,32,34,36,38,40,42,44,46,48), (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), (1,26,13,38)(2,43,14,31)(3,36,15,48)(4,29,16,41)(5,46,17,34)(6,39,18,27)(7,32,19,44)(8,25,20,37)(9,42,21,30)(10,35,22,47)(11,28,23,40)(12,45,24,33)>;
G:=Group( (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,3,5,7,9,11,13,15,17,19,21,23)(2,12,22,8,18,4,14,24,10,20,6,16)(25,35,45,31,41,27,37,47,33,43,29,39)(26,28,30,32,34,36,38,40,42,44,46,48), (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), (1,26,13,38)(2,43,14,31)(3,36,15,48)(4,29,16,41)(5,46,17,34)(6,39,18,27)(7,32,19,44)(8,25,20,37)(9,42,21,30)(10,35,22,47)(11,28,23,40)(12,45,24,33) );
G=PermutationGroup([[(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48)], [(1,3,5,7,9,11,13,15,17,19,21,23),(2,12,22,8,18,4,14,24,10,20,6,16),(25,35,45,31,41,27,37,47,33,43,29,39),(26,28,30,32,34,36,38,40,42,44,46,48)], [(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)], [(1,26,13,38),(2,43,14,31),(3,36,15,48),(4,29,16,41),(5,46,17,34),(6,39,18,27),(7,32,19,44),(8,25,20,37),(9,42,21,30),(10,35,22,47),(11,28,23,40),(12,45,24,33)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | ··· | 12V | 12W | 12X | 12Y | 12Z | 24A | ··· | 24P | 24Q | ··· | 24AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | 1 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C8 | S3×C12 | S3×C24 | S32 | C6.D6 | C3×S32 | C12.29D6 | C3×C6.D6 | C3×C12.29D6 |
| kernel | C3×C12.29D6 | C32×C3⋊C8 | C12×C3⋊S3 | C12.29D6 | C3×C3⋊Dic3 | C6×C3⋊S3 | C3×C3⋊C8 | C4×C3⋊S3 | C3×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊S3 | C3×C3⋊C8 | C3×C12 | C3⋊C8 | C3×C6 | C12 | C32 | C6 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×C12.29D6 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 46 |
| 0 | 0 | 27 | 0 |
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 0 | 22 | 0 |
| 72 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,27,0,0,46,0],[72,72,0,0,1,0,0,0,0,0,0,22,0,0,22,0],[72,72,0,0,0,1,0,0,0,0,0,46,0,0,46,0] >;
C3×C12.29D6 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{29}D_6 % in TeX
G:=Group("C3xC12.29D6"); // GroupNames label
G:=SmallGroup(432,415);
// by ID
G=gap.SmallGroup(432,415);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,92,80,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=1,c^6=b^3,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^5>;
// generators/relations