direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8×M4(2), C42.289C23, C8⋊10(C2×Q8), (C8×Q8)⋊29C2, C4⋊C4○3M4(2), C4.40(C4×Q8), C8⋊4Q8⋊34C2, (C4×Q8).26C4, C4.67(C22×Q8), C22.17(C4×Q8), C4⋊C8.360C22, C42.214(C2×C4), (C2×C8).426C23, (C2×C4).660C24, (C4×C8).332C22, C4.32(C2×M4(2)), (C22×Q8).31C4, (C4×M4(2)).30C2, (C4×Q8).330C22, C8⋊C4.161C22, C2.21(Q8○M4(2)), C4⋊M4(2).38C2, (C2×C42).773C22, C23.227(C22×C4), C22.186(C23×C4), C2.14(C22×M4(2)), (C22×C4).1522C23, (C2×M4(2)).363C22, C2.27(C2×C4×Q8), (C2×C4⋊C4).74C4, (C2×C4×Q8).44C2, (C4×Q8)○(C2×M4(2)), C4⋊C4.224(C2×C4), C4.311(C2×C4○D4), (C2×C4).315(C2×Q8), (C2×Q8).226(C2×C4), (C2×C4).899(C4○D4), (C2×C4).474(C22×C4), (C22×C4).347(C2×C4), SmallGroup(128,1695)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
Subgroups: 244 in 194 conjugacy classes, 150 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C2×M4(2), C2×M4(2), C22×Q8, C4×M4(2), C4⋊M4(2), C8×Q8, C8⋊4Q8, C2×C4×Q8, Q8×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, M4(2), C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C2×M4(2), C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, C22×M4(2), Q8○M4(2), Q8×M4(2)
►On 64 points
(1 26 19 38)(2 27 20 39)(3 28 21 40)(4 29 22 33)(5 30 23 34)(6 31 24 35)(7 32 17 36)(8 25 18 37)(9 45 57 53)(10 46 58 54)(11 47 59 55)(12 48 60 56)(13 41 61 49)(14 42 62 50)(15 43 63 51)(16 44 64 52)
(1 55 19 47)(2 56 20 48)(3 49 21 41)(4 50 22 42)(5 51 23 43)(6 52 24 44)(7 53 17 45)(8 54 18 46)(9 32 57 36)(10 25 58 37)(11 26 59 38)(12 27 60 39)(13 28 61 40)(14 29 62 33)(15 30 63 34)(16 31 64 35)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)
G:=sub<Sym(64)| (1,26,19,38)(2,27,20,39)(3,28,21,40)(4,29,22,33)(5,30,23,34)(6,31,24,35)(7,32,17,36)(8,25,18,37)(9,45,57,53)(10,46,58,54)(11,47,59,55)(12,48,60,56)(13,41,61,49)(14,42,62,50)(15,43,63,51)(16,44,64,52), (1,55,19,47)(2,56,20,48)(3,49,21,41)(4,50,22,42)(5,51,23,43)(6,52,24,44)(7,53,17,45)(8,54,18,46)(9,32,57,36)(10,25,58,37)(11,26,59,38)(12,27,60,39)(13,28,61,40)(14,29,62,33)(15,30,63,34)(16,31,64,35), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)>;
G:=Group( (1,26,19,38)(2,27,20,39)(3,28,21,40)(4,29,22,33)(5,30,23,34)(6,31,24,35)(7,32,17,36)(8,25,18,37)(9,45,57,53)(10,46,58,54)(11,47,59,55)(12,48,60,56)(13,41,61,49)(14,42,62,50)(15,43,63,51)(16,44,64,52), (1,55,19,47)(2,56,20,48)(3,49,21,41)(4,50,22,42)(5,51,23,43)(6,52,24,44)(7,53,17,45)(8,54,18,46)(9,32,57,36)(10,25,58,37)(11,26,59,38)(12,27,60,39)(13,28,61,40)(14,29,62,33)(15,30,63,34)(16,31,64,35), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64) );
G=PermutationGroup([[(1,26,19,38),(2,27,20,39),(3,28,21,40),(4,29,22,33),(5,30,23,34),(6,31,24,35),(7,32,17,36),(8,25,18,37),(9,45,57,53),(10,46,58,54),(11,47,59,55),(12,48,60,56),(13,41,61,49),(14,42,62,50),(15,43,63,51),(16,44,64,52)], [(1,55,19,47),(2,56,20,48),(3,49,21,41),(4,50,22,42),(5,51,23,43),(6,52,24,44),(7,53,17,45),(8,54,18,46),(9,32,57,36),(10,25,58,37),(11,26,59,38),(12,27,60,39),(13,28,61,40),(14,29,62,33),(15,30,63,34),(16,31,64,35)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4R | 4S | ··· | 4X | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | C4○D4 | M4(2) | Q8○M4(2) |
| kernel | Q8×M4(2) | C4×M4(2) | C4⋊M4(2) | C8×Q8 | C8⋊4Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | M4(2) | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 2 | 6 | 1 | 6 | 8 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of Q8×M4(2) ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 10 | 16 |
| 3 | 15 | 0 | 0 |
| 11 | 14 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 3 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,10,0,0,10,16],[3,11,0,0,15,14,0,0,0,0,16,0,0,0,0,16],[1,3,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;
Q8×M4(2) in GAP, Magma, Sage, TeX
Q_8\times M_4(2)
% in TeX
G:=Group("Q8xM4(2)"); // GroupNames label
G:=SmallGroup(128,1695);
// by ID
G=gap.SmallGroup(128,1695);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,268,2019,521,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations