p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊6Q16, C42.487C23, C4.732- (1+4), D4○3(C2.D8), C4⋊C4.270D4, (C4×Q16)⋊14C2, C8⋊2Q8⋊18C2, (C8×D4).10C2, C4.29(C2×Q16), C4.Q16⋊14C2, (C2×D4).356D4, C8.76(C4○D4), C2.54(D4○D8), (C4×C8).88C22, D4⋊3Q8.5C2, C8.18D4⋊15C2, C22.6(C2×Q16), C4⋊C4.243C23, C4⋊C8.300C22, (C2×C8).197C23, (C2×C4).530C24, C22⋊C4.114D4, C23.479(C2×D4), C4⋊Q8.163C22, C2.83(D4⋊6D4), C2.20(C22×Q16), C2.D8.63C22, (C4×D4).343C22, (C4×Q8).173C22, (C2×Q8).236C23, C22⋊Q8.99C22, C23.48D4⋊10C2, C22⋊C8.186C22, (C22×C8).165C22, (C2×Q16).139C22, C22.790(C22×D4), (C22×C4).1162C23, Q8⋊C4.162C22, (C2×C2.D8)⋊30C2, C4.112(C2×C4○D4), (C2×C4).173(C2×D4), (C2×C4⋊C4).682C22, SmallGroup(128,2070)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 312 in 178 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×6], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C2.D8, C2.D8 [×8], C2×C4⋊C4 [×4], C4×D4, C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16, C2×C2.D8 [×2], C8×D4, C4×Q16, C8.18D4 [×2], C4.Q16 [×2], C23.48D4 [×4], C8⋊2Q8, D4⋊3Q8 [×2], D4⋊6Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×Q16 [×6], C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, C22×Q16, D4○D8, D4⋊6Q16
Generators and relations
G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
►On 64 points
(1 34 18 43)(2 35 19 44)(3 36 20 45)(4 37 21 46)(5 38 22 47)(6 39 23 48)(7 40 24 41)(8 33 17 42)(9 53 61 25)(10 54 62 26)(11 55 63 27)(12 56 64 28)(13 49 57 29)(14 50 58 30)(15 51 59 31)(16 52 60 32)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 60 21 64)(18 59 22 63)(19 58 23 62)(20 57 24 61)(25 45 29 41)(26 44 30 48)(27 43 31 47)(28 42 32 46)(33 52 37 56)(34 51 38 55)(35 50 39 54)(36 49 40 53)
G:=sub<Sym(64)| (1,34,18,43)(2,35,19,44)(3,36,20,45)(4,37,21,46)(5,38,22,47)(6,39,23,48)(7,40,24,41)(8,33,17,42)(9,53,61,25)(10,54,62,26)(11,55,63,27)(12,56,64,28)(13,49,57,29)(14,50,58,30)(15,51,59,31)(16,52,60,32), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,60,21,64)(18,59,22,63)(19,58,23,62)(20,57,24,61)(25,45,29,41)(26,44,30,48)(27,43,31,47)(28,42,32,46)(33,52,37,56)(34,51,38,55)(35,50,39,54)(36,49,40,53)>;
G:=Group( (1,34,18,43)(2,35,19,44)(3,36,20,45)(4,37,21,46)(5,38,22,47)(6,39,23,48)(7,40,24,41)(8,33,17,42)(9,53,61,25)(10,54,62,26)(11,55,63,27)(12,56,64,28)(13,49,57,29)(14,50,58,30)(15,51,59,31)(16,52,60,32), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,60,21,64)(18,59,22,63)(19,58,23,62)(20,57,24,61)(25,45,29,41)(26,44,30,48)(27,43,31,47)(28,42,32,46)(33,52,37,56)(34,51,38,55)(35,50,39,54)(36,49,40,53) );
G=PermutationGroup([(1,34,18,43),(2,35,19,44),(3,36,20,45),(4,37,21,46),(5,38,22,47),(6,39,23,48),(7,40,24,41),(8,33,17,42),(9,53,61,25),(10,54,62,26),(11,55,63,27),(12,56,64,28),(13,49,57,29),(14,50,58,30),(15,51,59,31),(16,52,60,32)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64)], [(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)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,60,21,64),(18,59,22,63),(19,58,23,62),(20,57,24,61),(25,45,29,41),(26,44,30,48),(27,43,31,47),(28,42,32,46),(33,52,37,56),(34,51,38,55),(35,50,39,54),(36,49,40,53)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 13 | 13 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,2,16],[3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,13,0,0,0,0,0,4,13,0,0,8,13] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 4L | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 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 | D4 | D4 | D4 | C4○D4 | Q16 | 2- (1+4) | D4○D8 |
| kernel | D4⋊6Q16 | C2×C2.D8 | C8×D4 | C4×Q16 | C8.18D4 | C4.Q16 | C23.48D4 | C8⋊2Q8 | D4⋊3Q8 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | D4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_6Q_{16} % in TeX
G:=Group("D4:6Q16"); // GroupNames label
G:=SmallGroup(128,2070);
// by ID
G=gap.SmallGroup(128,2070);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,436,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations