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G = C4.C4≀C2  order 128 = 27

9th non-split extension by C4 of C4≀C2 acting via C4≀C2/C42=C2

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

Aliases: C4.9C4≀C2, C16⋊C4⋊3C2, (C2×C8).23D4, C4⋊1D4.1C4, C42.1(C2×C4), C42.C2.1C4, C8⋊C4.83C22, C22.10(C4.D4), C42.29C22.2C2, C2.4(C42.C22), (C2×C4).56(C22⋊C4), SmallGroup(128,87)

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

Derived series
C1 — C42 — C4.C4≀C2
Chief series
C1 — C2 — C22 — C2×C4 — C2×C8 — C8⋊C4 — C42.29C22 — C4.C4≀C2
Lower central
C1 — C2 — C2×C4 — C42 — C4.C4≀C2
Upper central
C1 — C2 — C2×C4 — C8⋊C4 — C4.C4≀C2
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C8⋊C4 — C4.C4≀C2

Generators and relations for C4.C4≀C2
 G = < a,b,c,d | a4=b4=c2=1, d4=cac=a-1, dbd-1=ab=ba, ad=da, cbc=b-1, dcd-1=b-1c >

C1
C2
2C2
16C2
C22
C4
C4
4C4
8C4
8C22
8C22
8C22
C2×C4
2C8
2C8
2C2×C4
4C2×C4
4D4
4C23
4D4
8D4
8D4
C2×C8
C42
C2×C8
2C4⋊C4
2C16
2C2×D4
2C16
2C16
2C16
4C4⋊C4
4C2×D4
C8⋊C4
C4⋊1D4
C42.C2
2M5(2)
2D4⋊C4
2D4⋊C4
2M5(2)
C16⋊C4
C16⋊C4
C42.29C22
C4.C4≀C2

Character table of C4.C4≀C2

 class 12A2B2C4A4B4C4D8A8B8C8D16A16B16C16D16E16F16G16H
 size 1121622816444488888888
ρ111111111111111111111    trivial
ρ2111-1111-1111111-1-1-11-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-1111-11111-1-1111-11-1    linear of order 2
ρ5111-11111-1-1-1-1i-ii-iii-i-i    linear of order 4
ρ61111111-1-1-1-1-1i-i-ii-iii-i    linear of order 4
ρ7111-11111-1-1-1-1-ii-ii-i-iii    linear of order 4
ρ81111111-1-1-1-1-1-iii-ii-i-ii    linear of order 4
ρ9222022-20-22-2200000000    orthogonal lifted from D4
ρ10222022-202-22-200000000    orthogonal lifted from D4
ρ1122-20-22000-2i02i00-1-i-1+i1+i01-i0    complex lifted from C4≀C2
ρ1222-20-220002i0-2i00-1+i-1-i1-i01+i0    complex lifted from C4≀C2
ρ1322-20-220002i0-2i001-i1+i-1+i0-1-i0    complex lifted from C4≀C2
ρ1422-202-2002i0-2i01+i-1+i000-1-i01-i    complex lifted from C4≀C2
ρ1522-202-2002i0-2i0-1-i1-i0001+i0-1+i    complex lifted from C4≀C2
ρ1622-202-200-2i02i01-i-1-i000-1+i01+i    complex lifted from C4≀C2
ρ1722-20-22000-2i02i001+i1-i-1-i0-1+i0    complex lifted from C4≀C2
ρ1822-202-200-2i02i0-1+i1+i0001-i0-1-i    complex lifted from C4≀C2
ρ194440-4-400000000000000    orthogonal lifted from C4.D4
ρ208-8000000000000000000    orthogonal faithful

Permutation representations of C4.C4≀C2
►On 16 points - transitive group 16T355
Generators in S16
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)

(1 9)(2 6 10 14)(4 16 12 8)(5 13)

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

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8), (1,9)(2,6,10,14)(4,16,12,8)(5,13), (1,5)(2,14)(6,10)(7,15)(8,16)(9,13), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
 
G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8), (1,9)(2,6,10,14)(4,16,12,8)(5,13), (1,5)(2,14)(6,10)(7,15)(8,16)(9,13), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8)], [(1,9),(2,6,10,14),(4,16,12,8),(5,13)], [(1,5),(2,14),(6,10),(7,15),(8,16),(9,13)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,355);
 

Matrix representation of C4.C4≀C2 ►in GL8(ℤ)

01000000
-10000000
00010000
00-100000
00000-100
00001000
0000000-1
00000010
,
10000000
01000000
00-100000
000-10000
00000100
0000-1000
0000000-1
00000010
,
0-1000000
-10000000
00100000
000-10000
0000-1000
00000100
00000001
00000010
,
00001000
00000-100
0000000-1
000000-10
00100000
000-10000
-10000000
01000000

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

C4.C4≀C2 in GAP, Magma, Sage, TeX 

C_4.C_4\wr C_2
 
% in TeX
 
G:=Group("C4.C4wrC2");
 
// GroupNames label
 
G:=SmallGroup(128,87);
 
// by ID
 
G=gap.SmallGroup(128,87);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,184,1690,521,80,1411,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^2=1,d^4=c*a*c=a^-1,d*b*d^-1=a*b=b*a,a*d=d*a,c*b*c=b^-1,d*c*d^-1=b^-1*c>;
 
// generators/relations
 

Export

Subgroup lattice of C4.C4≀C2 in TeX
Character table of C4.C4≀C2 in TeX

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