metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.20C42, C23.14D12, (C2×C24)⋊2C4, (C2×C8)⋊2Dic3, (C4×Dic3)⋊4C4, C12.12(C4⋊C4), (C2×C12).12Q8, (C2×C4).6Dic6, C3⋊2(C4.9C42), (C2×C12).110D4, C4.25(C4×Dic3), (C22×C6).47D4, (C22×C4).79D6, (C2×M4(2)).8S3, C12.87(C22⋊C4), C4.12(Dic3⋊C4), C22.20(D6⋊C4), (C6×M4(2)).12C2, C4.18(C6.D4), C22.10(C4⋊Dic3), C2.16(C6.C42), C6.16(C2.C42), (C22×C12).126C22, C23.26D6.10C2, (C2×C6).40(C4⋊C4), (C2×C4).140(C4×S3), (C2×C12).63(C2×C4), (C2×C4).22(C3⋊D4), (C2×C4).77(C2×Dic3), (C2×C6).11(C22⋊C4), SmallGroup(192,116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.20C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a5, ac=ca, cbc-1=a9b >
Subgroups: 232 in 94 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, C4.9C42, C23.26D6, C6×M4(2), C12.20C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4.9C42, C6.C42, C12.20C42
►On 48 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)
(1 42 13 26)(2 47 14 31)(3 40 15 36)(4 45 16 29)(5 38 17 34)(6 43 18 27)(7 48 19 32)(8 41 20 25)(9 46 21 30)(10 39 22 35)(11 44 23 28)(12 37 24 33)
(1 48 16 29 7 42 22 35)(2 37 17 30 8 43 23 36)(3 38 18 31 9 44 24 25)(4 39 19 32 10 45 13 26)(5 40 20 33 11 46 14 27)(6 41 21 34 12 47 15 28)
G:=sub<Sym(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), (1,42,13,26)(2,47,14,31)(3,40,15,36)(4,45,16,29)(5,38,17,34)(6,43,18,27)(7,48,19,32)(8,41,20,25)(9,46,21,30)(10,39,22,35)(11,44,23,28)(12,37,24,33), (1,48,16,29,7,42,22,35)(2,37,17,30,8,43,23,36)(3,38,18,31,9,44,24,25)(4,39,19,32,10,45,13,26)(5,40,20,33,11,46,14,27)(6,41,21,34,12,47,15,28)>;
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), (1,42,13,26)(2,47,14,31)(3,40,15,36)(4,45,16,29)(5,38,17,34)(6,43,18,27)(7,48,19,32)(8,41,20,25)(9,46,21,30)(10,39,22,35)(11,44,23,28)(12,37,24,33), (1,48,16,29,7,42,22,35)(2,37,17,30,8,43,23,36)(3,38,18,31,9,44,24,25)(4,39,19,32,10,45,13,26)(5,40,20,33,11,46,14,27)(6,41,21,34,12,47,15,28) );
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)], [(1,42,13,26),(2,47,14,31),(3,40,15,36),(4,45,16,29),(5,38,17,34),(6,43,18,27),(7,48,19,32),(8,41,20,25),(9,46,21,30),(10,39,22,35),(11,44,23,28),(12,37,24,33)], [(1,48,16,29,7,42,22,35),(2,37,17,30,8,43,23,36),(3,38,18,31,9,44,24,25),(4,39,19,32,10,45,13,26),(5,40,20,33,11,46,14,27),(6,41,21,34,12,47,15,28)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | Dic6 | C4×S3 | C3⋊D4 | D12 | C4.9C42 | C12.20C42 |
| kernel | C12.20C42 | C23.26D6 | C6×M4(2) | C4×Dic3 | C2×C24 | C2×M4(2) | C2×C12 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of C12.20C42 ►in GL4(𝔽73) generated by
| 0 | 27 | 0 | 0 |
| 46 | 46 | 0 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 46 | 46 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 14 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
G:=sub<GL(4,GF(73))| [0,46,0,0,27,46,0,0,0,0,0,46,0,0,27,46],[0,0,14,66,0,0,7,59,0,46,0,0,46,0,0,0],[0,0,7,59,0,0,14,66,1,0,0,0,0,1,0,0] >;
C12.20C42 in GAP, Magma, Sage, TeX
C_{12}._{20}C_4^2 % in TeX
G:=Group("C12.20C4^2"); // GroupNames label
G:=SmallGroup(192,116);
// by ID
G=gap.SmallGroup(192,116);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^4=a^6,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations