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G = D4×Q16order 128 = 27

Direct product of D4 and Q16

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

Aliases: D4×Q16, C42.453C23, C4.1402+ 1+4, C2.70D42, C42(C2×Q16), (C4×Q16)⋊9C2, (C8×D4).8C2, C8.82(C2×D4), (D4×Q8).4C2, C4⋊C4.257D4, Q8.29(C2×D4), C222(C2×Q16), C4⋊Q1612C2, C42Q1613C2, (C2×D4).349D4, (C4×C8).81C22, C22⋊Q168C2, C22⋊C4.96D4, C2.43(Q8○D8), C8.18D410C2, C4⋊C4.225C23, C4⋊C8.295C22, (C2×C8).180C23, (C2×C4).484C24, (C22×Q16)⋊13C2, C4.100(C22×D4), C23.468(C2×D4), C4⋊Q8.139C22, C2.18(C22×Q16), (C4×D4).327C22, (C2×Q16).36C22, (C2×Q8).392C23, (C4×Q8).145C22, C2.D8.189C22, C22⋊Q8.68C22, C22⋊C8.183C22, (C22×C8).159C22, Q8⋊C4.10C22, C22.744(C22×D4), (C22×C4).1128C23, (C22×Q8).336C22, (C2×D4)(C2×Q16), (C2×C4).162(C2×D4), SmallGroup(128,2018)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4×Q16
C1C2C22C2×C4C2×Q8C22×Q8C22×Q16 — D4×Q16
C1C2C2×C4 — D4×Q16
C1C22C4×D4 — D4×Q16
C1C2C2C2×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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×13], C22, C22 [×4], C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×20], D4 [×4], Q8 [×4], Q8 [×18], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], Q16 [×4], Q16 [×14], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×6], C2×Q8 [×16], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×4], C4⋊Q8 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16, C2×Q16 [×8], C2×Q16 [×8], C22×Q8 [×4], C8×D4, C4×Q16, C22⋊Q16 [×4], C42Q16 [×2], C8.18D4 [×2], C4⋊Q16, D4×Q8 [×2], C22×Q16 [×2], D4×Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C24, C2×Q16 [×6], C22×D4 [×2], 2+ 1+4, D42, C22×Q16, Q8○D8, D4×Q16

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

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4K4L···4Q8A8B8C8D8E···8J
order1222222244444···44···488888···8
size1111222222224···48···822224···4

35 irreducible representations

dim1111111112222244
type+++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D4D4D4D4Q162+ 1+4Q8○D8
kernelD4×Q16C8×D4C4×Q16C22⋊Q16C42Q16C8.18D4C4⋊Q16D4×Q8C22×Q16C22⋊C4C4⋊C4Q16C2×D4D4C4C2
# reps1114221222141812

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

1000
0100
001615
0011
,
16000
01600
0010
001616
,
31400
3300
00160
00016
,
101600
16700
00160
00016
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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