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## G = D4⋊4Q16order 128 = 27

### 3rd semidirect product of D4 and Q16 acting via Q16/Q8=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — D4⋊4Q16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — D4×Q8 — D4⋊4Q16
 Lower central C1 — C22 — C42 — D4⋊4Q16
 Upper central C1 — C22 — C42 — D4⋊4Q16
 Jennings C1 — C22 — C22 — C42 — D4⋊4Q16

Generators and relations for D44Q16
G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >

Subgroups: 264 in 117 conjugacy classes, 38 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×8], C22, C22 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×12], D4 [×2], D4, Q8 [×2], Q8 [×9], C23, C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×3], Q16 [×2], C22×C4 [×3], C2×D4, C2×Q8, C2×Q8 [×8], C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C2.D8 [×3], C4×D4, C4×D4, C4×Q8, C22⋊Q8 [×3], C4⋊Q8 [×2], C4⋊Q8, C2×Q16, C22×Q8, D4⋊C8, Q8⋊C8, C4.10D8, C42Q16, D4⋊Q8, C82Q8, D4×Q8, D44Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], Q16 [×2], C2×D4 [×3], C22≀C2, C2×D8, C2×Q16, C8⋊C22, C8.C22, C22⋊D8, C22⋊Q16, D4.10D4, D44Q16

Character table of D44Q16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 8 8 8 16 4 4 4 4 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 linear of order 2 ρ7 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ9 2 2 2 2 0 0 -2 -2 2 2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 2 2 2 2 -2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 -2 -2 -2 0 0 2 0 0 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 2 2 -2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -2 -2 -2 -2 0 0 2 0 0 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 2 2 2 -2 -2 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 -√2 √2 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ16 2 -2 2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 -√2 √2 0 0 orthogonal lifted from D8 ρ17 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 √2 -√2 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ18 2 -2 2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 √2 -√2 0 0 orthogonal lifted from D8 ρ19 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 0 0 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ20 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 0 0 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ21 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 0 0 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ22 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 0 0 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ23 4 -4 4 -4 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ24 4 -4 -4 4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of D44Q16
On 64 points
Generators in S64
(1 50 63 18)(2 19 64 51)(3 52 57 20)(4 21 58 53)(5 54 59 22)(6 23 60 55)(7 56 61 24)(8 17 62 49)(9 47 34 28)(10 29 35 48)(11 41 36 30)(12 31 37 42)(13 43 38 32)(14 25 39 44)(15 45 40 26)(16 27 33 46)
(1 22)(2 60)(3 24)(4 62)(5 18)(6 64)(7 20)(8 58)(9 43)(10 14)(11 45)(12 16)(13 47)(15 41)(17 21)(19 23)(25 48)(26 36)(27 42)(28 38)(29 44)(30 40)(31 46)(32 34)(33 37)(35 39)(49 53)(50 59)(51 55)(52 61)(54 63)(56 57)
(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 32 5 28)(2 31 6 27)(3 30 7 26)(4 29 8 25)(9 50 13 54)(10 49 14 53)(11 56 15 52)(12 55 16 51)(17 39 21 35)(18 38 22 34)(19 37 23 33)(20 36 24 40)(41 61 45 57)(42 60 46 64)(43 59 47 63)(44 58 48 62)

G:=sub<Sym(64)| (1,50,63,18)(2,19,64,51)(3,52,57,20)(4,21,58,53)(5,54,59,22)(6,23,60,55)(7,56,61,24)(8,17,62,49)(9,47,34,28)(10,29,35,48)(11,41,36,30)(12,31,37,42)(13,43,38,32)(14,25,39,44)(15,45,40,26)(16,27,33,46), (1,22)(2,60)(3,24)(4,62)(5,18)(6,64)(7,20)(8,58)(9,43)(10,14)(11,45)(12,16)(13,47)(15,41)(17,21)(19,23)(25,48)(26,36)(27,42)(28,38)(29,44)(30,40)(31,46)(32,34)(33,37)(35,39)(49,53)(50,59)(51,55)(52,61)(54,63)(56,57), (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), (1,32,5,28)(2,31,6,27)(3,30,7,26)(4,29,8,25)(9,50,13,54)(10,49,14,53)(11,56,15,52)(12,55,16,51)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(41,61,45,57)(42,60,46,64)(43,59,47,63)(44,58,48,62)>;

G:=Group( (1,50,63,18)(2,19,64,51)(3,52,57,20)(4,21,58,53)(5,54,59,22)(6,23,60,55)(7,56,61,24)(8,17,62,49)(9,47,34,28)(10,29,35,48)(11,41,36,30)(12,31,37,42)(13,43,38,32)(14,25,39,44)(15,45,40,26)(16,27,33,46), (1,22)(2,60)(3,24)(4,62)(5,18)(6,64)(7,20)(8,58)(9,43)(10,14)(11,45)(12,16)(13,47)(15,41)(17,21)(19,23)(25,48)(26,36)(27,42)(28,38)(29,44)(30,40)(31,46)(32,34)(33,37)(35,39)(49,53)(50,59)(51,55)(52,61)(54,63)(56,57), (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), (1,32,5,28)(2,31,6,27)(3,30,7,26)(4,29,8,25)(9,50,13,54)(10,49,14,53)(11,56,15,52)(12,55,16,51)(17,39,21,35)(18,38,22,34)(19,37,23,33)(20,36,24,40)(41,61,45,57)(42,60,46,64)(43,59,47,63)(44,58,48,62) );

G=PermutationGroup([(1,50,63,18),(2,19,64,51),(3,52,57,20),(4,21,58,53),(5,54,59,22),(6,23,60,55),(7,56,61,24),(8,17,62,49),(9,47,34,28),(10,29,35,48),(11,41,36,30),(12,31,37,42),(13,43,38,32),(14,25,39,44),(15,45,40,26),(16,27,33,46)], [(1,22),(2,60),(3,24),(4,62),(5,18),(6,64),(7,20),(8,58),(9,43),(10,14),(11,45),(12,16),(13,47),(15,41),(17,21),(19,23),(25,48),(26,36),(27,42),(28,38),(29,44),(30,40),(31,46),(32,34),(33,37),(35,39),(49,53),(50,59),(51,55),(52,61),(54,63),(56,57)], [(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)], [(1,32,5,28),(2,31,6,27),(3,30,7,26),(4,29,8,25),(9,50,13,54),(10,49,14,53),(11,56,15,52),(12,55,16,51),(17,39,21,35),(18,38,22,34),(19,37,23,33),(20,36,24,40),(41,61,45,57),(42,60,46,64),(43,59,47,63),(44,58,48,62)])

Matrix representation of D44Q16 in 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 0 6 0 0 3 0
,
 7 16 0 0 16 10 0 0 0 0 16 0 0 0 0 16
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],[3,3,0,0,14,3,0,0,0,0,0,3,0,0,6,0],[7,16,0,0,16,10,0,0,0,0,16,0,0,0,0,16] >;

D44Q16 in GAP, Magma, Sage, TeX

D_4\rtimes_4Q_{16}
% in TeX

G:=Group("D4:4Q16");
// GroupNames label

G:=SmallGroup(128,381);
// by ID

G=gap.SmallGroup(128,381);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,456,422,352,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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