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## G = D4×Q16order 128 = 27

### Direct product of D4 and Q16

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for D4×Q16
G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 424 in 236 conjugacy classes, 104 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×C8, C2×Q16, C2×Q16, C2×Q16, C22×Q8, C8×D4, C4×Q16, C22⋊Q16, C42Q16, C8.18D4, C4⋊Q16, D4×Q8, C22×Q16, D4×Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C24, C2×Q16, C22×D4, 2+ 1+4, D42, C22×Q16, Q8○D8, D4×Q16

Smallest permutation representation of D4×Q16
On 64 points
Generators in S64
(1 34 19 48)(2 35 20 41)(3 36 21 42)(4 37 22 43)(5 38 23 44)(6 39 24 45)(7 40 17 46)(8 33 18 47)(9 50 58 25)(10 51 59 26)(11 52 60 27)(12 53 61 28)(13 54 62 29)(14 55 63 30)(15 56 64 31)(16 49 57 32)
(1 5)(2 6)(3 7)(4 8)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)(49 53)(50 54)(51 55)(52 56)
(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 30 5 26)(2 29 6 25)(3 28 7 32)(4 27 8 31)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(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)

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

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

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

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 D4 Q16 2+ 1+4 Q8○D8 kernel D4×Q16 C8×D4 C4×Q16 C22⋊Q16 C4⋊2Q16 C8.18D4 C4⋊Q16 D4×Q8 C22×Q16 C22⋊C4 C4⋊C4 Q16 C2×D4 D4 C4 C2 # reps 1 1 1 4 2 2 1 2 2 2 1 4 1 8 1 2

Matrix representation of D4×Q16 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 16 15 0 0 1 1
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 16 16
,
 3 14 0 0 3 3 0 0 0 0 16 0 0 0 0 16
,
 10 16 0 0 16 7 0 0 0 0 16 0 0 0 0 16
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,1,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[10,16,0,0,16,7,0,0,0,0,16,0,0,0,0,16] >;

D4×Q16 in GAP, Magma, Sage, TeX

D_4\times Q_{16}
% in TeX

G:=Group("D4xQ16");
// GroupNames label

G:=SmallGroup(128,2018);
// by ID

G=gap.SmallGroup(128,2018);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2019,346,248,2804,1411,375,172]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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