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G = D16⋊C4  order 128 = 27

The semidirect product of D16 and C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D16⋊C4, SD32⋊2C4, C42.12D4, M5(2).13C22, C16⋊(C2×C4), D8⋊(C2×C4), Q16⋊(C2×C4), C8.Q8⋊2C2, C16⋊C4⋊2C2, C4.34(C4×D4), (C2×C8).32D4, D8⋊C4⋊1C2, C8.26D4⋊1C2, D8⋊2C4⋊3C2, (C2×C8).2C23, M5(2)⋊C2⋊6C2, C8.26(C4○D4), C16⋊C22.1C2, C8.12(C22×C4), C4○D8.2C22, C4.96(C8⋊C22), C4.Q8.2C22, C8⋊C4.4C22, (C2×D8).44C22, C2.14(D8⋊C4), C8.C4.2C22, C22.23(C8⋊C22), (C2×C4).277(C2×D4), 2-Sylow(PGammaL(2,81)), Aut(D16), Hol(C16), SmallGroup(128,913)

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

Derived series
C1 — C8 — D16⋊C4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8⋊C4 — C8.26D4 — D16⋊C4
Lower central
C1 — C2 — C4 — C8 — D16⋊C4
Upper central
C1 — C2 — C2×C4 — C8⋊C4 — D16⋊C4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D16⋊C4

Generators and relations for D16⋊C4
 G = < a,b,c | a16=b2=c4=1, bab=a-1, cac-1=a5, bc=cb >

Subgroups: 196 in 77 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, D8, D8, SD16, Q16, C22×C4, C2×D4, C4○D4, C8⋊C4, D4⋊C4, C4≀C2, C4.Q8, C8.C4, M5(2), D16, SD32, C4×D4, C8○D4, C2×D8, C4○D8, C16⋊C4, D8⋊2C4, M5(2)⋊C2, C8.Q8, D8⋊C4, C8.26D4, C16⋊C22, D16⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C4×D4, C8⋊C22, D8⋊C4, D16⋊C4

Character table of D16⋊C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G8A8B8C8D8E8F16A16B16C16D
 size 11288822448884444888888
ρ111111111111111111111111    trivial
ρ2111-1-1111-1-11111-11-1-1-1-11-11    linear of order 2
ρ311111111-1-1-11-11-11-1-1-11-11-1    linear of order 2
ρ4111-1-111111-11-1111111-1-1-1-1    linear of order 2
ρ511111-111-1-1-1-1-11-11-111-11-11    linear of order 2
ρ6111-1-1-11111-1-1-11111-1-11111    linear of order 2
ρ711111-111111-111111-1-1-1-1-1-1    linear of order 2
ρ8111-1-1-111-1-11-111-11-1111-11-1    linear of order 2
ρ911-11-11-11-iii-1-i-1i1-ii-i1-i-1i    linear of order 4
ρ1011-1-111-11-ii-i-1i-1i1-ii-i-1i1-i    linear of order 4
ρ1111-11-1-1-11i-i-i1i-1-i1ii-i-1-i1i    linear of order 4
ρ1211-1-11-1-11i-ii1-i-1-i1ii-i1i-1-i    linear of order 4
ρ1311-1-111-11i-ii-1-i-1-i1i-ii-1-i1i    linear of order 4
ρ1411-11-11-11i-i-i-1i-1-i1i-ii1i-1-i    linear of order 4
ρ1511-1-11-1-11-ii-i1i-1i1-i-ii1-i-1i    linear of order 4
ρ1611-11-1-1-11-iii1-i-1i1-i-ii-1i1-i    linear of order 4
ρ1722200022-2-2000-22-22000000    orthogonal lifted from D4
ρ182220002222000-2-2-2-2000000    orthogonal lifted from D4
ρ1922-2000-222i-2i00022i-2-2i000000    complex lifted from C4○D4
ρ2022-2000-22-2i2i0002-2i-22i000000    complex lifted from C4○D4
ρ21444000-4-4000000000000000    orthogonal lifted from C8⋊C22
ρ2244-40004-4000000000000000    orthogonal lifted from C8⋊C22
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of D16⋊C4
►On 16 points - transitive group 16T256
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(1 7)(2 6)(3 5)(8 16)(9 15)(10 14)(11 13)

(1 13 9 5)(2 10)(3 7 11 15)(6 14)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,7)(2,6)(3,5)(8,16)(9,15)(10,14)(11,13), (1,13,9,5)(2,10)(3,7,11,15)(6,14)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,7)(2,6)(3,5)(8,16)(9,15)(10,14)(11,13), (1,13,9,5)(2,10)(3,7,11,15)(6,14) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,7),(2,6),(3,5),(8,16),(9,15),(10,14),(11,13)], [(1,13,9,5),(2,10),(3,7,11,15),(6,14)]])
 
G:=TransitiveGroup(16,256);
 

Matrix representation of D16⋊C4 ►in GL8(ℤ)

00001000
00000-100
0000000-1
000000-10
00010000
00100000
01000000
10000000
,
00010000
00100000
01000000
10000000
00001000
00000-100
0000000-1
000000-10
,
0-1000000
10000000
00010000
00-100000
0000-1000
00000-100
00000010
00000001

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

D16⋊C4 in GAP, Magma, Sage, TeX 

D_{16}\rtimes C_4
 
% in TeX
 
G:=Group("D16:C4");
 
// GroupNames label
 
G:=SmallGroup(128,913);
 
// by ID
 
G=gap.SmallGroup(128,913);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,758,604,1123,1466,521,136,1411,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^5,b*c=c*b>;
 
// generators/relations
 

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Character table of D16⋊C4 in TeX

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