p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊12D4, C42.452C23, C4.1392+ 1+4, C2.69D42, (C8×D4)⋊14C2, 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⋊C8.317C22, C4⋊C4.224C23, (C2×C8).345C23, (C4×C8).119C22, (C2×C4).483C24, C22⋊C4.194D4, (C22×Q16)⋊12C2, C23.467(C2×D4), (C2×D4).217C23, (C4×D4).326C22, (C2×D8).137C22, 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
Generators and relations for Q16⋊12D4
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 >
Subgroups: 472 in 241 conjugacy classes, 96 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C22×C8, C2×D8, C2×SD16, C2×Q16, C2×Q16, C2×Q16, C4○D8, C22×Q8, C2×C4○D4, C8×D4, C4×Q16, Q8⋊D4, D4.7D4, Q8.D4, C8⋊7D4, C8.18D4, C8.12D4, Q8⋊5D4, C22×Q16, C2×C4○D8, Q16⋊12D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4○D8, C22×D4, 2+ 1+4, D42, C2×C4○D8, Q8○D8, Q16⋊12D4
►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 35 5 39)(2 34 6 38)(3 33 7 37)(4 40 8 36)(9 32 13 28)(10 31 14 27)(11 30 15 26)(12 29 16 25)(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 11 19 48)(2 10 20 47)(3 9 21 46)(4 16 22 45)(5 15 23 44)(6 14 24 43)(7 13 17 42)(8 12 18 41)(25 56 58 36)(26 55 59 35)(27 54 60 34)(28 53 61 33)(29 52 62 40)(30 51 63 39)(31 50 64 38)(32 49 57 37)
(2 8)(3 7)(4 6)(9 42)(10 41)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 21)(18 20)(22 24)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(34 40)(35 39)(36 38)(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), (1,35,5,39)(2,34,6,38)(3,33,7,37)(4,40,8,36)(9,32,13,28)(10,31,14,27)(11,30,15,26)(12,29,16,25)(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), (1,11,19,48)(2,10,20,47)(3,9,21,46)(4,16,22,45)(5,15,23,44)(6,14,24,43)(7,13,17,42)(8,12,18,41)(25,56,58,36)(26,55,59,35)(27,54,60,34)(28,53,61,33)(29,52,62,40)(30,51,63,39)(31,50,64,38)(32,49,57,37), (2,8)(3,7)(4,6)(9,42)(10,41)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,21)(18,20)(22,24)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(34,40)(35,39)(36,38)(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), (1,35,5,39)(2,34,6,38)(3,33,7,37)(4,40,8,36)(9,32,13,28)(10,31,14,27)(11,30,15,26)(12,29,16,25)(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), (1,11,19,48)(2,10,20,47)(3,9,21,46)(4,16,22,45)(5,15,23,44)(6,14,24,43)(7,13,17,42)(8,12,18,41)(25,56,58,36)(26,55,59,35)(27,54,60,34)(28,53,61,33)(29,52,62,40)(30,51,63,39)(31,50,64,38)(32,49,57,37), (2,8)(3,7)(4,6)(9,42)(10,41)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,21)(18,20)(22,24)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(34,40)(35,39)(36,38)(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)], [(1,35,5,39),(2,34,6,38),(3,33,7,37),(4,40,8,36),(9,32,13,28),(10,31,14,27),(11,30,15,26),(12,29,16,25),(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)], [(1,11,19,48),(2,10,20,47),(3,9,21,46),(4,16,22,45),(5,15,23,44),(6,14,24,43),(7,13,17,42),(8,12,18,41),(25,56,58,36),(26,55,59,35),(27,54,60,34),(28,53,61,33),(29,52,62,40),(30,51,63,39),(31,50,64,38),(32,49,57,37)], [(2,8),(3,7),(4,6),(9,42),(10,41),(11,48),(12,47),(13,46),(14,45),(15,44),(16,43),(17,21),(18,20),(22,24),(25,64),(26,63),(27,62),(28,61),(29,60),(30,59),(31,58),(32,57),(34,40),(35,39),(36,38),(50,56),(51,55),(52,54)]])
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 |
Matrix representation of Q16⋊12D4 ►in 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] >;
Q16⋊12D4 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