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G = D16⋊3C4  order 128 = 27

2nd semidirect product of D16 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 5), monomial

Aliases: D16⋊3C4, Q32⋊3C4, C16.23D4, C8.4SD16, C4.8SD32, M6(2)⋊5C2, C22.3D16, C16.4(C2×C4), C16⋊4C4⋊1C2, (C2×C4).12D8, (C2×C8).84D4, C4○D16.2C2, C2.9(C2.D16), C8.16(C22⋊C4), (C2×C16).13C22, C4.16(D4⋊C4), SmallGroup(128,150)

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

Derived series
C1 — C16 — D16⋊3C4
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C4○D16 — D16⋊3C4
Lower central
C1 — C2 — C4 — C8 — C16 — D16⋊3C4
Upper central
C1 — C2 — C2×C4 — C2×C8 — C2×C16 — D16⋊3C4
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C2×C16 — D16⋊3C4

Generators and relations for D16⋊3C4
 G = < a,b,c | a16=b2=c4=1, bab=a-1, cac-1=a7, cbc-1=a13b >

C1
C2
2C2
16C2
C22
C4
C4
8C22
8C4
16C4
C8
C8
C2×C4
4D4
4Q8
8D4
8C2×C4
8C2×C4
C16
C16
C2×C8
2D8
2Q16
4C4○D4
4SD16
4C4⋊C4
C2×C16
Q32
D16
2C32
2SD32
2C4○D8
2C2.D8
C16⋊4C4
M6(2)
C4○D16
D16⋊3C4

Character table of D16⋊3C4

 class 12A2B2C4A4B4C4D4E8A8B8C16A16B16C16D16E16F32A32B32C32D32E32F32G32H
 size 112162216161622422224444444444
ρ111111111111111111111111111    trivial
ρ2111111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111-111-1-1-111111111111111111    linear of order 2
ρ4111-11111-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-111-1i-i-111-1-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ611-111-1-ii-111-1-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ711-1-11-1i-i111-1-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ811-1-11-1-ii111-1-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ9222022000222-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1022-202-200022-22222-2-200000000    orthogonal lifted from D4
ρ112220-2-2000000-√2√2-√2√2√2-√2ζ167-ζ16ζ167-ζ16-ζ165+ζ163-ζ165+ζ163-ζ167+ζ16ζ165-ζ163-ζ167+ζ16ζ165-ζ163    orthogonal lifted from D16
ρ12222022000-2-2-2000000√2√2-√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ13222022000-2-2-2000000-√2-√2√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ142220-2-2000000√2-√2√2-√2-√2√2-ζ165+ζ163-ζ165+ζ163-ζ167+ζ16-ζ167+ζ16ζ165-ζ163ζ167-ζ16ζ165-ζ163ζ167-ζ16    orthogonal lifted from D16
ρ152220-2-2000000-√2√2-√2√2√2-√2-ζ167+ζ16-ζ167+ζ16ζ165-ζ163ζ165-ζ163ζ167-ζ16-ζ165+ζ163ζ167-ζ16-ζ165+ζ163    orthogonal lifted from D16
ρ162220-2-2000000√2-√2√2-√2-√2√2ζ165-ζ163ζ165-ζ163ζ167-ζ16ζ167-ζ16-ζ165+ζ163-ζ167+ζ16-ζ165+ζ163-ζ167+ζ16    orthogonal lifted from D16
ρ1722-202-2000-2-22000000√-2-√-2-√-2√-2√-2√-2-√-2-√-2    complex lifted from SD16
ρ1822-20-22000000-√2√2-√2√2-√2√2ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ1922-202-2000-2-22000000-√-2√-2√-2-√-2-√-2-√-2√-2√-2    complex lifted from SD16
ρ2022-20-22000000-√2√2-√2√2-√2√2ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ2122-20-22000000√2-√2√2-√2√2-√2ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ2222-20-22000000√2-√2√2-√2√2-√2ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ234-400000002√2-2√202ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ1690000000000    complex faithful
ρ244-40000000-2√22√202ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ16110000000000    complex faithful
ρ254-40000000-2√22√202ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ1630000000000    complex faithful
ρ264-400000002√2-2√202ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ160000000000    complex faithful

Smallest permutation representation of D16⋊3C4
►On 32 points
Generators in S32
(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)

(1 21)(2 20)(3 19)(4 18)(5 17)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)

(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 20 25 28)(18 27 26 19)(21 32 29 24)(22 23 30 31)
 
G:=sub<Sym(32)| (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,20,25,28)(18,27,26,19)(21,32,29,24)(22,23,30,31)>;
 
G:=Group( (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,20,25,28)(18,27,26,19)(21,32,29,24)(22,23,30,31) );
 
G=PermutationGroup([[(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,20,25,28),(18,27,26,19),(21,32,29,24),(22,23,30,31)]])
 

Matrix representation of D16⋊3C4 ►in GL4(𝔽7) generated by

0003
1006
1021
3635
,
5603
4222
2421
5035
,
1321
1042
1451
3241
G:=sub<GL(4,GF(7))| [0,1,1,3,0,0,0,6,0,0,2,3,3,6,1,5],[5,4,2,5,6,2,4,0,0,2,2,3,3,2,1,5],[1,1,1,3,3,0,4,2,2,4,5,4,1,2,1,1] >;
 

D16⋊3C4 in GAP, Magma, Sage, TeX 

D_{16}\rtimes_3C_4
 
% in TeX
 
G:=Group("D16:3C4");
 
// GroupNames label
 
G:=SmallGroup(128,150);
 
// by ID
 
G=gap.SmallGroup(128,150);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,891,604,1018,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^13*b>;
 
// generators/relations
 

Export

Subgroup lattice of D16⋊3C4 in TeX
Character table of D16⋊3C4 in TeX

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