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G = D4xQ16order 128 = 27

Direct product of D4 and Q16

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

Aliases: D4xQ16, C42.453C23, C4.1402+ 1+4, C2.70D42, C4:2(C2xQ16), (C4xQ16):9C2, (C8xD4).8C2, C8.82(C2xD4), (D4xQ8).4C2, C4:C4.257D4, Q8.29(C2xD4), C22:2(C2xQ16), C4:Q16:12C2, C4:2Q16:13C2, (C2xD4).349D4, (C4xC8).81C22, C22:Q16:8C2, C22:C4.96D4, C2.43(Q8oD8), C8.18D4:10C2, C4:C4.225C23, C4:C8.295C22, (C2xC8).180C23, (C2xC4).484C24, (C22xQ16):13C2, C4.100(C22xD4), C23.468(C2xD4), C4:Q8.139C22, C2.18(C22xQ16), (C4xD4).327C22, (C2xQ16).36C22, (C2xQ8).392C23, (C4xQ8).145C22, C2.D8.189C22, C22:Q8.68C22, C22:C8.183C22, (C22xC8).159C22, Q8:C4.10C22, C22.744(C22xD4), (C22xC4).1128C23, (C22xQ8).336C22, (C2xD4)o(C2xQ16), (C2xC4).162(C2xD4), SmallGroup(128,2018)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — D4xQ16
C1C2C22C2xC4C2xQ8C22xQ8C22xQ16 — D4xQ16
C1C2C2xC4 — D4xQ16
C1C22C4xD4 — D4xQ16
C1C2C2C2xC4 — D4xQ16

Generators and relations for D4xQ16
 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, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, Q16, Q16, C22xC4, C22xC4, C2xD4, C2xQ8, C2xQ8, C4xC8, C22:C8, Q8:C4, C4:C8, C2.D8, C4xD4, C4xD4, C4xQ8, C22:Q8, C22:Q8, C4:Q8, C4:Q8, C22xC8, C2xQ16, C2xQ16, C2xQ16, C22xQ8, C8xD4, C4xQ16, C22:Q16, C4:2Q16, C8.18D4, C4:Q16, D4xQ8, C22xQ16, D4xQ16
Quotients: C1, C2, C22, D4, C23, Q16, C2xD4, C24, C2xQ16, C22xD4, 2+ 1+4, D42, C22xQ16, Q8oD8, D4xQ16

Smallest permutation representation of D4xQ16
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 2A2B2C2D2E2F2G4A4B4C4D4E···4K4L···4Q8A8B8C8D8E···8J
order1222222244444···44···488888···8
size1111222222224···48···822224···4

35 irreducible representations

dim1111111112222244
type+++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D4D4D4D4Q162+ 1+4Q8oD8
kernelD4xQ16C8xD4C4xQ16C22:Q16C4:2Q16C8.18D4C4:Q16D4xQ8C22xQ16C22:C4C4:C4Q16C2xD4D4C4C2
# reps1114221222141812

Matrix representation of D4xQ16 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] >;

D4xQ16 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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