metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊6M4(2), (C2×C8)⋊20D6, D6⋊C8⋊37C2, C4.88(C2×D12), (C2×C24)⋊35C22, C4.38(D6⋊C4), C12.306(C2×D4), (C2×C4).151D12, (C2×C12).170D4, (C2×M4(2))⋊6S3, (S3×C23).7C4, C23.59(C4×S3), (C6×M4(2))⋊16C2, C3⋊2(C24.4C4), (C22×C4).365D6, C2.20(S3×M4(2)), C6.31(C2×M4(2)), C12.25(C22⋊C4), (C2×C12).866C23, C22.27(D6⋊C4), (C22×Dic3).16C4, (C22×C12).185C22, (S3×C2×C4).9C4, (C2×C3⋊C8)⋊28C22, C2.27(C2×D6⋊C4), (S3×C22×C4).2C2, (C2×C4).159(C4×S3), C4.132(C2×C3⋊D4), C6.55(C2×C22⋊C4), C22.145(S3×C2×C4), (C2×C12).105(C2×C4), (S3×C2×C4).284C22, (C2×C4.Dic3)⋊14C2, (C22×C6).66(C2×C4), (C2×C4).195(C3⋊D4), (C2×C6).19(C22⋊C4), (C22×S3).63(C2×C4), (C2×C4).808(C22×S3), (C2×C6).136(C22×C4), (C2×Dic3).102(C2×C4), SmallGroup(192,685)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊6M4(2)
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c5 >
Subgroups: 504 in 190 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22⋊C8, C2×M4(2), C2×M4(2), C23×C4, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C24.4C4, D6⋊C8, C2×C4.Dic3, C6×M4(2), S3×C22×C4, D6⋊6M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, M4(2), C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C2×M4(2), D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C24.4C4, S3×M4(2), C2×D6⋊C4, D6⋊6M4(2)
►On 48 points
(1 25 47 16 23 36)(2 26 48 9 24 37)(3 27 41 10 17 38)(4 28 42 11 18 39)(5 29 43 12 19 40)(6 30 44 13 20 33)(7 31 45 14 21 34)(8 32 46 15 22 35)
(1 36)(2 48)(3 38)(4 42)(5 40)(6 44)(7 34)(8 46)(9 37)(10 41)(11 39)(12 43)(13 33)(14 45)(15 35)(16 47)(17 27)(19 29)(21 31)(23 25)
(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)
G:=sub<Sym(48)| (1,25,47,16,23,36)(2,26,48,9,24,37)(3,27,41,10,17,38)(4,28,42,11,18,39)(5,29,43,12,19,40)(6,30,44,13,20,33)(7,31,45,14,21,34)(8,32,46,15,22,35), (1,36)(2,48)(3,38)(4,42)(5,40)(6,44)(7,34)(8,46)(9,37)(10,41)(11,39)(12,43)(13,33)(14,45)(15,35)(16,47)(17,27)(19,29)(21,31)(23,25), (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)>;
G:=Group( (1,25,47,16,23,36)(2,26,48,9,24,37)(3,27,41,10,17,38)(4,28,42,11,18,39)(5,29,43,12,19,40)(6,30,44,13,20,33)(7,31,45,14,21,34)(8,32,46,15,22,35), (1,36)(2,48)(3,38)(4,42)(5,40)(6,44)(7,34)(8,46)(9,37)(10,41)(11,39)(12,43)(13,33)(14,45)(15,35)(16,47)(17,27)(19,29)(21,31)(23,25), (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) );
G=PermutationGroup([[(1,25,47,16,23,36),(2,26,48,9,24,37),(3,27,41,10,17,38),(4,28,42,11,18,39),(5,29,43,12,19,40),(6,30,44,13,20,33),(7,31,45,14,21,34),(8,32,46,15,22,35)], [(1,36),(2,48),(3,38),(4,42),(5,40),(6,44),(7,34),(8,46),(9,37),(10,41),(11,39),(12,43),(13,33),(14,45),(15,35),(16,47),(17,27),(19,29),(21,31),(23,25)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | M4(2) | C4×S3 | D12 | C3⋊D4 | C4×S3 | S3×M4(2) |
| kernel | D6⋊6M4(2) | D6⋊C8 | C2×C4.Dic3 | C6×M4(2) | S3×C22×C4 | S3×C2×C4 | C22×Dic3 | S3×C23 | C2×M4(2) | C2×C12 | C2×C8 | C22×C4 | D6 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 4 | 2 | 1 | 8 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of D6⋊6M4(2) ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 27 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 30 | 13 |
| 0 | 0 | 60 | 43 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,1],[72,0,0,0,0,1,0,0,0,0,72,0,0,0,1,1],[0,72,0,0,27,0,0,0,0,0,30,60,0,0,13,43],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;
D6⋊6M4(2) in GAP, Magma, Sage, TeX
D_6\rtimes_6M_4(2)
% in TeX
G:=Group("D6:6M4(2)"); // GroupNames label
G:=SmallGroup(192,685);
// by ID
G=gap.SmallGroup(192,685);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^5>;
// generators/relations