metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.8C42, C42⋊8Dic3, C6.7C4≀C2, (C4×C12)⋊12C4, C4⋊Dic3⋊5C4, C4.Dic3⋊1C4, (C2×C42).8S3, C12.27(C4⋊C4), (C2×C12).58Q8, C4.8(C4×Dic3), C4.44(D6⋊C4), C3⋊1(C42⋊6C4), (C2×C12).478D4, (C2×C4).162D12, (C2×C4).42Dic6, C4.20(C4⋊Dic3), (C22×C4).428D6, (C22×C6).176D4, C2.3(C42⋊4S3), C12.59(C22⋊C4), C4.22(Dic3⋊C4), C22.36(D6⋊C4), C23.79(C3⋊D4), C6.1(C2.C42), C2.3(C6.C42), C23.26D6.1C2, C22.12(Dic3⋊C4), (C22×C12).532C22, C22.8(C6.D4), (C2×C4×C12).15C2, (C2×C4).98(C4×S3), (C2×C6).33(C4⋊C4), (C2×C12).217(C2×C4), (C2×C4).71(C2×Dic3), (C2×C4.Dic3).1C2, (C2×C4).231(C3⋊D4), (C2×C6).49(C22⋊C4), SmallGroup(192,82)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.8C42
G = < a,b,c | a12=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a3b >
Subgroups: 232 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C4×C12, C22×C12, C22×C12, C42⋊6C4, C2×C4.Dic3, C23.26D6, C2×C4×C12, C12.8C42
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≀C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C42⋊6C4, C42⋊4S3, C6.C42, C12.8C42
►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 37 23 25)(2 48 24 36)(3 47 13 35)(4 46 14 34)(5 45 15 33)(6 44 16 32)(7 43 17 31)(8 42 18 30)(9 41 19 29)(10 40 20 28)(11 39 21 27)(12 38 22 26)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)
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), (1,37,23,25)(2,48,24,36)(3,47,13,35)(4,46,14,34)(5,45,15,33)(6,44,16,32)(7,43,17,31)(8,42,18,30)(9,41,19,29)(10,40,20,28)(11,39,21,27)(12,38,22,26), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,37,34,43)(29,38,35,44)(30,39,36,45)>;
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), (1,37,23,25)(2,48,24,36)(3,47,13,35)(4,46,14,34)(5,45,15,33)(6,44,16,32)(7,43,17,31)(8,42,18,30)(9,41,19,29)(10,40,20,28)(11,39,21,27)(12,38,22,26), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,37,34,43)(29,38,35,44)(30,39,36,45) );
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)], [(1,37,23,25),(2,48,24,36),(3,47,13,35),(4,46,14,34),(5,45,15,33),(6,44,16,32),(7,43,17,31),(8,42,18,30),(9,41,19,29),(10,40,20,28),(11,39,21,27),(12,38,22,26)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,37,34,43),(29,38,35,44),(30,39,36,45)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | C4≀C2 | C42⋊4S3 |
| kernel | C12.8C42 | C2×C4.Dic3 | C23.26D6 | C2×C4×C12 | C4.Dic3 | C4⋊Dic3 | C4×C12 | C2×C42 | C2×C12 | C2×C12 | C22×C6 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 16 |
Matrix representation of C12.8C42 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 70 |
| 46 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 27 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 27 |
G:=sub<GL(3,GF(73))| [1,0,0,0,24,0,0,0,70],[46,0,0,0,0,1,0,1,0],[27,0,0,0,72,0,0,0,27] >;
C12.8C42 in GAP, Magma, Sage, TeX
C_{12}._8C_4^2 % in TeX
G:=Group("C12.8C4^2"); // GroupNames label
G:=SmallGroup(192,82);
// by ID
G=gap.SmallGroup(192,82);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,1123,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations