p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8.4M4(2), C42.303C23, C4.1192- (1+4), (C8×Q8)⋊33C2, C8⋊4Q8⋊40C2, Q8○3(C22⋊C8), (C4×Q8).32C4, C4⋊C8.368C22, (C2×C8).441C23, (C2×C4).681C24, C42.229(C2×C4), (C4×C8).340C22, C4.36(C2×M4(2)), (C22×Q8).34C4, C22.17(C8○D4), (C4×Q8).334C22, C8⋊C4.100C22, C22⋊C8.237C22, C42.6C4.33C2, (C2×C42).788C22, C22.204(C23×C4), C23.232(C22×C4), C2.22(C22×M4(2)), C42.12C4.46C2, (C22×C4).1285C23, C2.24(C23.32C23), (C2×C4⋊C4).81C4, (C2×C4×Q8).47C2, C2.32(C2×C8○D4), C4⋊C4.232(C2×C4), (C2×Q8).212(C2×C4), (C22×C4).360(C2×C4), (C2×C4).279(C22×C4), SmallGroup(128,1716)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 228 in 182 conjugacy classes, 136 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], Q8 [×4], Q8 [×6], C23, C42 [×12], C4⋊C4 [×12], C2×C8 [×8], C22×C4, C22×C4 [×6], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4×C8 [×6], C8⋊C4 [×6], C22⋊C8, C22⋊C8 [×3], C4⋊C8 [×12], C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×2], C4×Q8 [×6], C22×Q8, C42.12C4 [×3], C42.6C4 [×3], C8×Q8 [×2], C8⋊4Q8 [×6], C2×C4×Q8, Q8.4M4(2)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C8○D4 [×2], C23×C4, 2- (1+4) [×2], C23.32C23, C22×M4(2), C2×C8○D4, Q8.4M4(2)
Generators and relations
G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2c5 >
►On 64 points
(1 52 61 22)(2 23 62 53)(3 54 63 24)(4 17 64 55)(5 56 57 18)(6 19 58 49)(7 50 59 20)(8 21 60 51)(9 46 26 35)(10 36 27 47)(11 48 28 37)(12 38 29 41)(13 42 30 39)(14 40 31 43)(15 44 32 33)(16 34 25 45)
(1 36 61 47)(2 37 62 48)(3 38 63 41)(4 39 64 42)(5 40 57 43)(6 33 58 44)(7 34 59 45)(8 35 60 46)(9 21 26 51)(10 22 27 52)(11 23 28 53)(12 24 29 54)(13 17 30 55)(14 18 31 56)(15 19 32 49)(16 20 25 50)
(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 36 61 47)(2 44 62 33)(3 38 63 41)(4 46 64 35)(5 40 57 43)(6 48 58 37)(7 34 59 45)(8 42 60 39)(9 55 26 17)(10 22 27 52)(11 49 28 19)(12 24 29 54)(13 51 30 21)(14 18 31 56)(15 53 32 23)(16 20 25 50)
G:=sub<Sym(64)| (1,52,61,22)(2,23,62,53)(3,54,63,24)(4,17,64,55)(5,56,57,18)(6,19,58,49)(7,50,59,20)(8,21,60,51)(9,46,26,35)(10,36,27,47)(11,48,28,37)(12,38,29,41)(13,42,30,39)(14,40,31,43)(15,44,32,33)(16,34,25,45), (1,36,61,47)(2,37,62,48)(3,38,63,41)(4,39,64,42)(5,40,57,43)(6,33,58,44)(7,34,59,45)(8,35,60,46)(9,21,26,51)(10,22,27,52)(11,23,28,53)(12,24,29,54)(13,17,30,55)(14,18,31,56)(15,19,32,49)(16,20,25,50), (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), (1,36,61,47)(2,44,62,33)(3,38,63,41)(4,46,64,35)(5,40,57,43)(6,48,58,37)(7,34,59,45)(8,42,60,39)(9,55,26,17)(10,22,27,52)(11,49,28,19)(12,24,29,54)(13,51,30,21)(14,18,31,56)(15,53,32,23)(16,20,25,50)>;
G:=Group( (1,52,61,22)(2,23,62,53)(3,54,63,24)(4,17,64,55)(5,56,57,18)(6,19,58,49)(7,50,59,20)(8,21,60,51)(9,46,26,35)(10,36,27,47)(11,48,28,37)(12,38,29,41)(13,42,30,39)(14,40,31,43)(15,44,32,33)(16,34,25,45), (1,36,61,47)(2,37,62,48)(3,38,63,41)(4,39,64,42)(5,40,57,43)(6,33,58,44)(7,34,59,45)(8,35,60,46)(9,21,26,51)(10,22,27,52)(11,23,28,53)(12,24,29,54)(13,17,30,55)(14,18,31,56)(15,19,32,49)(16,20,25,50), (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), (1,36,61,47)(2,44,62,33)(3,38,63,41)(4,46,64,35)(5,40,57,43)(6,48,58,37)(7,34,59,45)(8,42,60,39)(9,55,26,17)(10,22,27,52)(11,49,28,19)(12,24,29,54)(13,51,30,21)(14,18,31,56)(15,53,32,23)(16,20,25,50) );
G=PermutationGroup([(1,52,61,22),(2,23,62,53),(3,54,63,24),(4,17,64,55),(5,56,57,18),(6,19,58,49),(7,50,59,20),(8,21,60,51),(9,46,26,35),(10,36,27,47),(11,48,28,37),(12,38,29,41),(13,42,30,39),(14,40,31,43),(15,44,32,33),(16,34,25,45)], [(1,36,61,47),(2,37,62,48),(3,38,63,41),(4,39,64,42),(5,40,57,43),(6,33,58,44),(7,34,59,45),(8,35,60,46),(9,21,26,51),(10,22,27,52),(11,23,28,53),(12,24,29,54),(13,17,30,55),(14,18,31,56),(15,19,32,49),(16,20,25,50)], [(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)], [(1,36,61,47),(2,44,62,33),(3,38,63,41),(4,46,64,35),(5,40,57,43),(6,48,58,37),(7,34,59,45),(8,42,60,39),(9,55,26,17),(10,22,27,52),(11,49,28,19),(12,24,29,54),(13,51,30,21),(14,18,31,56),(15,53,32,23),(16,20,25,50)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 8 | 16 |
| 0 | 0 | 9 | 9 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,13,0,0,0,0,0,16,0,0,0,0,16],[0,9,0,0,9,0,0,0,0,0,8,9,0,0,16,9],[0,13,0,0,13,0,0,0,0,0,16,1,0,0,0,1] >;
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 | 4 |
| type | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | C8○D4 | 2- (1+4) |
| kernel | Q8.4M4(2) | C42.12C4 | C42.6C4 | C8×Q8 | C8⋊4Q8 | C2×C4×Q8 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | Q8 | C22 | C4 |
| # reps | 1 | 3 | 3 | 2 | 6 | 1 | 6 | 8 | 2 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
Q_8._4M_{4(2)} % in TeX
G:=Group("Q8.4M4(2)"); // GroupNames label
G:=SmallGroup(128,1716);
// by ID
G=gap.SmallGroup(128,1716);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,891,100,675,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^5>;
// generators/relations