p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).20D4, C42.378C23, C8.5(C2×D4), C4⋊C4.242D4, C8.2D4⋊6C2, (C2×C8).146D4, C4⋊Q16⋊19C2, C22⋊C4.82D4, C2.22(Q8○D8), C4.14(C22×D4), C8.12D4⋊17C2, C4.43(C4⋊1D4), (C4×C8).176C22, (C2×C4).354C24, (C2×C8).561C23, (C22×Q16)⋊19C2, C23.456(C2×D4), C4⋊Q8.111C22, C8○2M4(2)⋊14C2, (C2×D8).131C22, (C2×D4).120C23, (C2×Q8).108C23, C8⋊C4.119C22, C22.20(C4⋊1D4), (C22×C8).273C22, (C2×Q16).127C22, (C2×SD16).21C22, C4.4D4.33C22, C22.614(C22×D4), (C22×C4).1044C23, (C22×Q8).312C22, C23.38C23⋊13C2, C42⋊C2.327C22, (C2×M4(2)).274C22, (C2×C4○D8).19C2, (C2×C4).140(C2×D4), C2.33(C2×C4⋊1D4), (C2×C8.C22)⋊25C2, (C2×C4○D4).160C22, SmallGroup(128,1888)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 492 in 266 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×8], Q8 [×16], C23, C23 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], D8 [×2], SD16 [×12], Q16 [×18], C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×10], C4○D4 [×8], C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×4], C4⋊Q8 [×4], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×6], C2×Q16, C2×Q16 [×8], C2×Q16 [×4], C4○D8 [×4], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4 [×2], C8○2M4(2), C4⋊Q16 [×2], C8.12D4 [×2], C8.2D4 [×4], C23.38C23 [×2], C22×Q16, C2×C4○D8, C2×C8.C22 [×2], M4(2).20D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C22×D4 [×3], C2×C4⋊1D4, Q8○D8 [×2], M4(2).20D4
Generators and relations
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, ac=ca, dad-1=a3, bc=cb, bd=db, dcd-1=a4c3 >
►On 64 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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)(33 46)(34 43)(35 48)(36 45)(37 42)(38 47)(39 44)(40 41)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)
(1 20 63 42 5 24 59 46)(2 21 64 43 6 17 60 47)(3 22 57 44 7 18 61 48)(4 23 58 45 8 19 62 41)(9 50 34 26 13 54 38 30)(10 51 35 27 14 55 39 31)(11 52 36 28 15 56 40 32)(12 53 37 29 16 49 33 25)
(1 8 5 4)(2 3 6 7)(9 35 13 39)(10 38 14 34)(11 33 15 37)(12 36 16 40)(17 44 21 48)(18 47 22 43)(19 42 23 46)(20 45 24 41)(25 28 29 32)(26 31 30 27)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)
G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,20,63,42,5,24,59,46)(2,21,64,43,6,17,60,47)(3,22,57,44,7,18,61,48)(4,23,58,45,8,19,62,41)(9,50,34,26,13,54,38,30)(10,51,35,27,14,55,39,31)(11,52,36,28,15,56,40,32)(12,53,37,29,16,49,33,25), (1,8,5,4)(2,3,6,7)(9,35,13,39)(10,38,14,34)(11,33,15,37)(12,36,16,40)(17,44,21,48)(18,47,22,43)(19,42,23,46)(20,45,24,41)(25,28,29,32)(26,31,30,27)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63)>;
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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)(33,46)(34,43)(35,48)(36,45)(37,42)(38,47)(39,44)(40,41)(49,59)(50,64)(51,61)(52,58)(53,63)(54,60)(55,57)(56,62), (1,20,63,42,5,24,59,46)(2,21,64,43,6,17,60,47)(3,22,57,44,7,18,61,48)(4,23,58,45,8,19,62,41)(9,50,34,26,13,54,38,30)(10,51,35,27,14,55,39,31)(11,52,36,28,15,56,40,32)(12,53,37,29,16,49,33,25), (1,8,5,4)(2,3,6,7)(9,35,13,39)(10,38,14,34)(11,33,15,37)(12,36,16,40)(17,44,21,48)(18,47,22,43)(19,42,23,46)(20,45,24,41)(25,28,29,32)(26,31,30,27)(49,56,53,52)(50,51,54,55)(57,64,61,60)(58,59,62,63) );
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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,25),(2,30),(3,27),(4,32),(5,29),(6,26),(7,31),(8,28),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24),(33,46),(34,43),(35,48),(36,45),(37,42),(38,47),(39,44),(40,41),(49,59),(50,64),(51,61),(52,58),(53,63),(54,60),(55,57),(56,62)], [(1,20,63,42,5,24,59,46),(2,21,64,43,6,17,60,47),(3,22,57,44,7,18,61,48),(4,23,58,45,8,19,62,41),(9,50,34,26,13,54,38,30),(10,51,35,27,14,55,39,31),(11,52,36,28,15,56,40,32),(12,53,37,29,16,49,33,25)], [(1,8,5,4),(2,3,6,7),(9,35,13,39),(10,38,14,34),(11,33,15,37),(12,36,16,40),(17,44,21,48),(18,47,22,43),(19,42,23,46),(20,45,24,41),(25,28,29,32),(26,31,30,27),(49,56,53,52),(50,51,54,55),(57,64,61,60),(58,59,62,63)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 12 | 5 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,14,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,12,12,0,0,0,0,12,5] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | Q8○D8 |
| kernel | M4(2).20D4 | C8○2M4(2) | C4⋊Q16 | C8.12D4 | C8.2D4 | C23.38C23 | C22×Q16 | C2×C4○D8 | C2×C8.C22 | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}._{20}D_4 % in TeX
G:=Group("M4(2).20D4"); // GroupNames label
G:=SmallGroup(128,1888);
// by ID
G=gap.SmallGroup(128,1888);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,184,521,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^3>;
// generators/relations