Copied to
clipboard

G = D4.Q16  order 128 = 27

2nd non-split extension by D4 of Q16 acting via Q16/C8=C2

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

Aliases: D4.2Q16, C42.234C23, (C8×D4).7C2, C8⋊1C8⋊10C2, C4⋊C4.209D4, (C2×C8).317D4, C4.26(C2×Q16), (C2×D4).195D4, C4.44(C4○D8), (C4×C8).59C22, D4⋊2Q8.9C2, D4⋊Q8.6C2, C4.6Q16⋊5C2, C4.SD16⋊9C2, C4⋊Q8.57C22, C4.10D8⋊26C2, C4⋊C8.182C22, C4.92(C8⋊C22), (C4×D4).283C22, C2.14(D4.3D4), C2.11(D4.2D4), C2.10(C8.18D4), C22.195(C4⋊D4), (C2×C4).19(C4○D4), (C2×C4).1269(C2×D4), SmallGroup(128,415)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — D4.Q16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — C8×D4 — D4.Q16
Lower central
C1 — C22 — C42 — D4.Q16
Upper central
C1 — C22 — C42 — D4.Q16
Jennings
C1 — C22 — C22 — C42 — D4.Q16

Generators and relations for D4.Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 176 in 80 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4⋊Q8, C22×C8, C4.10D8, C4.6Q16, C8⋊1C8, C8×D4, D4⋊Q8, D4⋊2Q8, C4.SD16, D4.Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C4⋊D4, C2×Q16, C4○D8, C8⋊C22, D4.2D4, C8.18D4, D4.3D4, D4.Q16

Character table of D4.Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111442222444161622224444448888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111-11-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ31111-1-11111-11-1-1-11111-1-1-111-11111    linear of order 2
ρ41111111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ511111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ61111-1-11111-11-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ71111-1-11111-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ81111111111111-1-11111111111-1-1-1-1    linear of order 2
ρ92222-2-2-22-222-220000000000000000    orthogonal lifted from D4
ρ102222002-22-20-20002222000-2-200000    orthogonal lifted from D4
ρ112222002-22-20-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ12222222-22-22-2-2-20000000000000000    orthogonal lifted from D4
ρ1322-2-22-2020-200000√2-√2-√2√2-√2√2√2-√2√2-√20000    symplectic lifted from Q16, Schur index 2
ρ1422-2-22-2020-200000-√2√2√2-√2√2-√2-√2√2-√2√20000    symplectic lifted from Q16, Schur index 2
ρ1522-2-2-22020-200000√2-√2-√2√2√2-√2-√2-√2√2√20000    symplectic lifted from Q16, Schur index 2
ρ1622-2-2-22020-200000-√2√2√2-√2-√2√2√2√2-√2-√20000    symplectic lifted from Q16, Schur index 2
ρ17222200-2-2-2-2020000000-2i-2i2i002i0000    complex lifted from C4○D4
ρ182-22-200-2020000002i-2i2i-2i000000-√-2-√2√-2√2    complex lifted from C4○D8
ρ19222200-2-2-2-20200000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ2022-2-2000-2022i0-2i00-√2√2√2-√2-√-2√-2-√-2-√2√2√-20000    complex lifted from C4○D8
ρ212-22-200-202000000-2i2i-2i2i000000-√-2√2√-2-√2    complex lifted from C4○D8
ρ222-22-200-202000000-2i2i-2i2i000000√-2-√2-√-2√2    complex lifted from C4○D8
ρ232-22-200-2020000002i-2i2i-2i000000√-2√2-√-2-√2    complex lifted from C4○D8
ρ2422-2-2000-202-2i02i00√2-√2-√2√2-√-2√-2-√-2√2-√2√-20000    complex lifted from C4○D8
ρ2522-2-2000-2022i0-2i00√2-√2-√2√2√-2-√-2√-2√2-√2-√-20000    complex lifted from C4○D8
ρ2622-2-2000-202-2i02i00-√2√2√2-√2√-2-√-2√-2-√2√2-√-20000    complex lifted from C4○D8
ρ274-44-40040-400000000000000000000    orthogonal lifted from C8⋊C22
ρ284-4-44000000000002√-22√-2-2√-2-2√-20000000000    complex lifted from D4.3D4
ρ294-4-4400000000000-2√-2-2√-22√-22√-20000000000    complex lifted from D4.3D4

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

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

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

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

16000
01600
0001
00160
,
1000
01600
0001
0010
,
2000
0900
00130
00013
,
0100
16000
00143
0033
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[2,0,0,0,0,9,0,0,0,0,13,0,0,0,0,13],[0,16,0,0,1,0,0,0,0,0,14,3,0,0,3,3] >;
 

D4.Q16 in GAP, Magma, Sage, TeX 

D_4.Q_{16}
 
% in TeX
 
G:=Group("D4.Q16");
 
// GroupNames label
 
G:=SmallGroup(128,415);
 
// by ID
 
G=gap.SmallGroup(128,415);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,736,422,1123,136,2804,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of D4.Q16 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁