p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊12D4, C42.452C23, C4.1392+ (1+4), (C8×D4)⋊14C2, C2.69(D42), C8.81(C2×D4), C22⋊C4○2Q16, C8⋊7D4⋊10C2, C4⋊C4.256D4, (C4×Q16)⋊24C2, Q8⋊5D4⋊7C2, Q8.28(C2×D4), Q8⋊D4⋊33C2, D4.7D4⋊8C2, (C2×D4).231D4, C8.12D4⋊9C2, C22⋊4(C4○D8), Q8.D4⋊6C2, C2.42(Q8○D8), C4.99(C22×D4), C8.18D4⋊22C2, C4⋊C4.224C23, C4⋊C8.317C22, (C4×C8).119C22, (C2×C8).345C23, (C2×C4).483C24, C22⋊C4.194D4, (C22×Q16)⋊12C2, C23.467(C2×D4), (C2×D8).137C22, (C4×D4).326C22, (C2×D4).217C23, C4⋊D4.68C22, (C4×Q8).144C22, (C2×Q8).391C23, C2.D8.188C22, C22⋊Q8.67C22, D4⋊C4.11C22, C22⋊C8.222C22, (C22×C8).194C22, (C2×Q16).170C22, (C2×SD16).95C22, C4.4D4.56C22, C22.743(C22×D4), (C22×C4).1127C23, Q8⋊C4.179C22, (C22×Q8).335C22, C22⋊C4○(C2×Q16), (C2×C4○D8)⋊13C2, C2.56(C2×C4○D8), (C2×C4).921(C2×D4), (C2×C4○D4).194C22, SmallGroup(128,2017)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 472 in 241 conjugacy classes, 96 normal (44 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×18], D4 [×14], Q8 [×4], Q8 [×10], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], D8 [×2], SD16 [×6], Q16 [×4], Q16 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C4.4D4 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×2], C2×Q16 [×2], C2×Q16 [×4], C4○D8 [×4], C22×Q8 [×2], C2×C4○D4 [×2], C8×D4, C4×Q16, Q8⋊D4 [×2], D4.7D4 [×2], Q8.D4 [×2], C8⋊7D4, C8.18D4, C8.12D4, Q8⋊5D4 [×2], C22×Q16, C2×C4○D8, Q16⋊12D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C4○D8 [×2], C22×D4 [×2], 2+ (1+4), D42, C2×C4○D8, Q8○D8, Q16⋊12D4
Generators and relations
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=dad=a-1, cbc-1=dbd=a4b, dcd=c-1 >
►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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 36 13 40)(10 35 14 39)(11 34 15 38)(12 33 16 37)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 16 19 48)(2 15 20 47)(3 14 21 46)(4 13 22 45)(5 12 23 44)(6 11 24 43)(7 10 17 42)(8 9 18 41)(25 36 52 62)(26 35 53 61)(27 34 54 60)(28 33 55 59)(29 40 56 58)(30 39 49 57)(31 38 50 64)(32 37 51 63)
(2 8)(3 7)(4 6)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 48)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 64)(50 56)(51 55)(52 54)
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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63), (1,16,19,48)(2,15,20,47)(3,14,21,46)(4,13,22,45)(5,12,23,44)(6,11,24,43)(7,10,17,42)(8,9,18,41)(25,36,52,62)(26,35,53,61)(27,34,54,60)(28,33,55,59)(29,40,56,58)(30,39,49,57)(31,38,50,64)(32,37,51,63), (2,8)(3,7)(4,6)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,48)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,64)(50,56)(51,55)(52,54)>;
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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63), (1,16,19,48)(2,15,20,47)(3,14,21,46)(4,13,22,45)(5,12,23,44)(6,11,24,43)(7,10,17,42)(8,9,18,41)(25,36,52,62)(26,35,53,61)(27,34,54,60)(28,33,55,59)(29,40,56,58)(30,39,49,57)(31,38,50,64)(32,37,51,63), (2,8)(3,7)(4,6)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,48)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,64)(50,56)(51,55)(52,54) );
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)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,36,13,40),(10,35,14,39),(11,34,15,38),(12,33,16,37),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63)], [(1,16,19,48),(2,15,20,47),(3,14,21,46),(4,13,22,45),(5,12,23,44),(6,11,24,43),(7,10,17,42),(8,9,18,41),(25,36,52,62),(26,35,53,61),(27,34,54,60),(28,33,55,59),(29,40,56,58),(30,39,49,57),(31,38,50,64),(32,37,51,63)], [(2,8),(3,7),(4,6),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,48),(17,21),(18,20),(22,24),(25,27),(28,32),(29,31),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,64),(50,56),(51,55),(52,54)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 3 | 11 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 4 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,8,4],[0,1,0,0,16,0,0,0,0,0,16,16,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | C4○D8 | 2+ (1+4) | Q8○D8 |
| kernel | Q16⋊12D4 | C8×D4 | C4×Q16 | Q8⋊D4 | D4.7D4 | Q8.D4 | C8⋊7D4 | C8.18D4 | C8.12D4 | Q8⋊5D4 | C22×Q16 | C2×C4○D8 | C22⋊C4 | C4⋊C4 | Q16 | C2×D4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_{16}\rtimes_{12}D_4 % in TeX
G:=Group("Q16:12D4"); // GroupNames label
G:=SmallGroup(128,2017);
// by ID
G=gap.SmallGroup(128,2017);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,346,248,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations