metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊5M4(2), (C2×C8)⋊18D6, D6⋊C8⋊22C2, C22⋊C8⋊11S3, (C4×S3).46D4, C4.195(S3×D4), (C2×C6)⋊1M4(2), (C2×C24)⋊26C22, C12.354(C2×D4), (S3×C23).5C4, C23.52(C4×S3), C6.5(C2×M4(2)), C22⋊4(C8⋊S3), C3⋊1(C24.4C4), (C22×C4).320D6, C2.12(S3×M4(2)), C12.55D4⋊23C2, (C2×C12).821C23, D6.10(C22⋊C4), (C22×Dic3).10C4, Dic3.11(C22⋊C4), (C22×C12).338C22, (S3×C2×C4).18C4, C2.9(C2×C8⋊S3), (C2×C3⋊C8)⋊27C22, (C2×C8⋊S3)⋊11C2, C6.8(C2×C22⋊C4), C2.9(S3×C22⋊C4), (C3×C22⋊C8)⋊20C2, (C2×C4).132(C4×S3), (S3×C22×C4).17C2, C22.103(S3×C2×C4), (C2×C12).154(C2×C4), (S3×C2×C4).273C22, (C22×C6).39(C2×C4), (C2×C6).76(C22×C4), (C22×S3).55(C2×C4), (C2×C4).763(C22×S3), (C2×Dic3).85(C2×C4), SmallGroup(192,285)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊M4(2)
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a4b, dcd=c5 >
Subgroups: 512 in 190 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C23, Dic3, 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, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22⋊C8, C22⋊C8, C2×M4(2), C23×C4, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C24.4C4, D6⋊C8, C12.55D4, C3×C22⋊C8, C2×C8⋊S3, S3×C22×C4, D6⋊M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, M4(2), C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C2×M4(2), C8⋊S3, S3×C2×C4, S3×D4, C24.4C4, S3×C22⋊C4, C2×C8⋊S3, S3×M4(2), D6⋊M4(2)
►On 48 points
(1 24 35 12 46 29)(2 30 47 13 36 17)(3 18 37 14 48 31)(4 32 41 15 38 19)(5 20 39 16 42 25)(6 26 43 9 40 21)(7 22 33 10 44 27)(8 28 45 11 34 23)
(1 35)(2 17)(3 37)(4 19)(5 39)(6 21)(7 33)(8 23)(9 43)(10 27)(11 45)(12 29)(13 47)(14 31)(15 41)(16 25)(26 40)(28 34)(30 36)(32 38)
(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 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 36)(18 33)(19 38)(20 35)(21 40)(22 37)(23 34)(24 39)(25 46)(26 43)(27 48)(28 45)(29 42)(30 47)(31 44)(32 41)
G:=sub<Sym(48)| (1,24,35,12,46,29)(2,30,47,13,36,17)(3,18,37,14,48,31)(4,32,41,15,38,19)(5,20,39,16,42,25)(6,26,43,9,40,21)(7,22,33,10,44,27)(8,28,45,11,34,23), (1,35)(2,17)(3,37)(4,19)(5,39)(6,21)(7,33)(8,23)(9,43)(10,27)(11,45)(12,29)(13,47)(14,31)(15,41)(16,25)(26,40)(28,34)(30,36)(32,38), (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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,36)(18,33)(19,38)(20,35)(21,40)(22,37)(23,34)(24,39)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41)>;
G:=Group( (1,24,35,12,46,29)(2,30,47,13,36,17)(3,18,37,14,48,31)(4,32,41,15,38,19)(5,20,39,16,42,25)(6,26,43,9,40,21)(7,22,33,10,44,27)(8,28,45,11,34,23), (1,35)(2,17)(3,37)(4,19)(5,39)(6,21)(7,33)(8,23)(9,43)(10,27)(11,45)(12,29)(13,47)(14,31)(15,41)(16,25)(26,40)(28,34)(30,36)(32,38), (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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,36)(18,33)(19,38)(20,35)(21,40)(22,37)(23,34)(24,39)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41) );
G=PermutationGroup([[(1,24,35,12,46,29),(2,30,47,13,36,17),(3,18,37,14,48,31),(4,32,41,15,38,19),(5,20,39,16,42,25),(6,26,43,9,40,21),(7,22,33,10,44,27),(8,28,45,11,34,23)], [(1,35),(2,17),(3,37),(4,19),(5,39),(6,21),(7,33),(8,23),(9,43),(10,27),(11,45),(12,29),(13,47),(14,31),(15,41),(16,25),(26,40),(28,34),(30,36),(32,38)], [(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)], [(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11),(17,36),(18,33),(19,38),(20,35),(21,40),(22,37),(23,34),(24,39),(25,46),(26,43),(27,48),(28,45),(29,42),(30,47),(31,44),(32,41)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | M4(2) | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | S3×D4 | S3×M4(2) |
| kernel | D6⋊M4(2) | D6⋊C8 | C12.55D4 | C3×C22⋊C8 | C2×C8⋊S3 | S3×C22×C4 | S3×C2×C4 | C22×Dic3 | S3×C23 | C22⋊C8 | C4×S3 | C2×C8 | C22×C4 | D6 | C2×C6 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 1 | 4 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of D6⋊M4(2) ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
| 16 | 65 | 0 | 0 |
| 8 | 57 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,72,72,0,0,0,0,1,72,0,0,0,72],[16,8,0,0,65,57,0,0,0,0,1,0,0,0,2,72],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,0,72] >;
D6⋊M4(2) in GAP, Magma, Sage, TeX
D_6\rtimes M_4(2)
% in TeX
G:=Group("D6:M4(2)"); // GroupNames label
G:=SmallGroup(192,285);
// by ID
G=gap.SmallGroup(192,285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^4*b,d*c*d=c^5>;
// generators/relations