metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).19D6, (C4×S3).35D4, C4.147(S3×D4), C12.90(C2×D4), C4.D4⋊6S3, (C2×D4).122D6, C12.D4⋊4C2, (S3×M4(2))⋊6C2, C23.12(C4×S3), (C2×C12).2C23, C12.47D4⋊9C2, D6.2(C22⋊C4), (C6×D4).12C22, (C22×Dic3).4C4, C4.Dic3.1C22, (C2×Dic6).44C22, Dic3.16(C22⋊C4), (C3×M4(2)).17C22, C3⋊1(M4(2).8C22), (C2×C3⋊D4).2C4, (S3×C2×C4).2C22, C22.15(S3×C2×C4), C2.14(S3×C22⋊C4), C6.13(C2×C22⋊C4), (C2×C6).9(C22×C4), (C2×C4).2(C22×S3), (C22×C6).7(C2×C4), (C2×D4⋊2S3).2C2, (C3×C4.D4)⋊10C2, (C2×Dic3).2(C2×C4), (C22×S3).15(C2×C4), SmallGroup(192,304)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).19D6
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >
Subgroups: 400 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4.D4, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, M4(2).8C22, C12.47D4, C12.D4, C3×C4.D4, S3×M4(2), C2×D4⋊2S3, M4(2).19D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4, M4(2).8C22, S3×C22⋊C4, M4(2).19D6
►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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)
(1 38 45 27 19 12)(2 35 46 32 20 9)(3 36 47 25 21 10)(4 33 48 30 22 15)(5 34 41 31 23 16)(6 39 42 28 24 13)(7 40 43 29 17 14)(8 37 44 26 18 11)
(1 12 5 16)(2 13 6 9)(3 10 7 14)(4 11 8 15)(17 40 21 36)(18 33 22 37)(19 38 23 34)(20 39 24 35)(25 43 29 47)(26 44 30 48)(27 41 31 45)(28 42 32 46)
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48), (1,38,45,27,19,12)(2,35,46,32,20,9)(3,36,47,25,21,10)(4,33,48,30,22,15)(5,34,41,31,23,16)(6,39,42,28,24,13)(7,40,43,29,17,14)(8,37,44,26,18,11), (1,12,5,16)(2,13,6,9)(3,10,7,14)(4,11,8,15)(17,40,21,36)(18,33,22,37)(19,38,23,34)(20,39,24,35)(25,43,29,47)(26,44,30,48)(27,41,31,45)(28,42,32,46)>;
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(42,46)(44,48), (1,38,45,27,19,12)(2,35,46,32,20,9)(3,36,47,25,21,10)(4,33,48,30,22,15)(5,34,41,31,23,16)(6,39,42,28,24,13)(7,40,43,29,17,14)(8,37,44,26,18,11), (1,12,5,16)(2,13,6,9)(3,10,7,14)(4,11,8,15)(17,40,21,36)(18,33,22,37)(19,38,23,34)(20,39,24,35)(25,43,29,47)(26,44,30,48)(27,41,31,45)(28,42,32,46) );
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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32),(33,37),(35,39),(42,46),(44,48)], [(1,38,45,27,19,12),(2,35,46,32,20,9),(3,36,47,25,21,10),(4,33,48,30,22,15),(5,34,41,31,23,16),(6,39,42,28,24,13),(7,40,43,29,17,14),(8,37,44,26,18,11)], [(1,12,5,16),(2,13,6,9),(3,10,7,14),(4,11,8,15),(17,40,21,36),(18,33,22,37),(19,38,23,34),(20,39,24,35),(25,43,29,47),(26,44,30,48),(27,41,31,45),(28,42,32,46)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 6 | 12 | 12 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | S3×D4 | M4(2).8C22 | M4(2).19D6 |
| kernel | M4(2).19D6 | C12.47D4 | C12.D4 | C3×C4.D4 | S3×M4(2) | C2×D4⋊2S3 | C22×Dic3 | C2×C3⋊D4 | C4.D4 | C4×S3 | M4(2) | C2×D4 | C23 | C4 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 1 |
Matrix representation of M4(2).19D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 72 | 46 | 71 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 59 | 0 | 13 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 46 | 0 | 46 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 46 | 72 | 46 | 71 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 27 | 1 | 27 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 46 | 72 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,59,0,0,72,0,1,0,0,0,46,46,0,13,0,0,71,0,0,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,46,0,0,0,72,0,0,0,0,0,0,1,46,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,46,1,0,0,0,0,72,0,0,0,0,1,46,0,0,0,0,0,71,0,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,27,1,46,0,0,0,1,0,72,0,0,72,27,0,0,0,0,0,2,0,72] >;
M4(2).19D6 in GAP, Magma, Sage, TeX
M_4(2)._{19}D_6 % in TeX
G:=Group("M4(2).19D6"); // GroupNames label
G:=SmallGroup(192,304);
// by ID
G=gap.SmallGroup(192,304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,219,58,570,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^5*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations