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G = C4.4D4⋊C4  order 128 = 27

6th semidirect product of C4.4D4 and C4 acting faithfully

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

Aliases: C4.4D4⋊6C4, C42⋊C4⋊4C2, (C2×D4).133D4, (C22×D4)⋊12C4, C23⋊C4⋊4C22, C42.10(C2×C4), (C22×C4).95D4, C23.11(C2×D4), C42⋊C2⋊10C4, C4.18(C23⋊C4), (C2×D4).22C23, C4⋊1D4.54C22, C23.37(C22⋊C4), C22.29C24.8C2, C23.C23⋊15C2, (C2×D4).38(C2×C4), C2.40(C2×C23⋊C4), (C2×C4).97(C22×C4), (C22×C4).32(C2×C4), (C2×Q8).107(C2×C4), (C2×C4).27(C22⋊C4), (C2×C4○D4).75C22, C22.64(C2×C22⋊C4), SmallGroup(128,860)

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

Derived series
C1 — C2×C4 — C4.4D4⋊C4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C2×C4○D4 — C22.29C24 — C4.4D4⋊C4
Lower central
C1 — C2 — C22 — C2×C4 — C4.4D4⋊C4
Upper central
C1 — C2 — C2×C4 — C2×C4○D4 — C4.4D4⋊C4
Jennings
C1 — C2 — C22 — C2×D4 — C4.4D4⋊C4

Generators and relations for C4.4D4⋊C4
 G = < a,b,c,d | a4=b4=d4=1, c2=a2, dbd-1=ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b-1, dcd-1=a2c >

Subgroups: 396 in 135 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C23⋊C4, C23⋊C4, C42⋊C2, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C42⋊C4, C23.C23, C22.29C24, C4.4D4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C4.4D4⋊C4

Character table of C4.4D4⋊C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 11244488224448888888888
ρ111111111111111111111111    trivial
ρ211111-1-11-1-1-1-111-111-1-111-1-1    linear of order 2
ρ311111-1-11-1-1-1-111-1-1-111-1-111    linear of order 2
ρ4111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111-11-1-1-1-1-11-11-1-11-111-11    linear of order 2
ρ6111111-1-111111-1-1-1-1-11111-1    linear of order 2
ρ7111111-1-111111-1-1111-1-1-1-11    linear of order 2
ρ811111-11-1-1-1-1-11-1111-11-1-11-1    linear of order 2
ρ9111-1-1-1-11-1-1111-11i-iii-ii-i-i    linear of order 4
ρ10111-1-111111-1-11-1-1i-i-i-i-iiii    linear of order 4
ρ11111-1-1-1-11-1-1111-11-ii-i-ii-iii    linear of order 4
ρ12111-1-111111-1-11-1-1-iiiii-i-i-i    linear of order 4
ρ13111-1-1-11-1-1-11111-1-ii-ii-ii-ii    linear of order 4
ρ14111-1-11-1-111-1-1111-iii-i-iii-i    linear of order 4
ρ15111-1-1-11-1-1-11111-1i-ii-ii-ii-i    linear of order 4
ρ16111-1-11-1-111-1-1111i-i-iii-i-ii    linear of order 4
ρ17222-22200-2-22-2-20000000000    orthogonal lifted from D4
ρ182222-2-200222-2-20000000000    orthogonal lifted from D4
ρ19222-22-20022-22-20000000000    orthogonal lifted from D4
ρ202222-2200-2-2-22-20000000000    orthogonal lifted from D4
ρ2144-400000-440000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000004-40000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C4.4D4⋊C4
►On 16 points - transitive group 16T244
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(5 7)(6 8)(9 12 11 10)(13 14 15 16)

(1 6 3 8)(2 5 4 7)(9 16 11 14)(10 15 12 13)

(1 10 2 9)(3 12 4 11)(5 14 6 13)(7 16 8 15)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,7)(6,8)(9,12,11,10)(13,14,15,16), (1,6,3,8)(2,5,4,7)(9,16,11,14)(10,15,12,13), (1,10,2,9)(3,12,4,11)(5,14,6,13)(7,16,8,15)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,7)(6,8)(9,12,11,10)(13,14,15,16), (1,6,3,8)(2,5,4,7)(9,16,11,14)(10,15,12,13), (1,10,2,9)(3,12,4,11)(5,14,6,13)(7,16,8,15) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(5,7),(6,8),(9,12,11,10),(13,14,15,16)], [(1,6,3,8),(2,5,4,7),(9,16,11,14),(10,15,12,13)], [(1,10,2,9),(3,12,4,11),(5,14,6,13),(7,16,8,15)]])
 
G:=TransitiveGroup(16,244);
 

►On 16 points - transitive group 16T263
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 10)(2 11)(3 12)(4 9)(5 14 7 16)(6 15 8 13)

(1 10 3 12)(2 9 4 11)(5 13 7 15)(6 16 8 14)

(1 5 11 14)(2 8 10 15)(3 7 9 16)(4 6 12 13)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11)(3,12)(4,9)(5,14,7,16)(6,15,8,13), (1,10,3,12)(2,9,4,11)(5,13,7,15)(6,16,8,14), (1,5,11,14)(2,8,10,15)(3,7,9,16)(4,6,12,13)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11)(3,12)(4,9)(5,14,7,16)(6,15,8,13), (1,10,3,12)(2,9,4,11)(5,13,7,15)(6,16,8,14), (1,5,11,14)(2,8,10,15)(3,7,9,16)(4,6,12,13) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10),(2,11),(3,12),(4,9),(5,14,7,16),(6,15,8,13)], [(1,10,3,12),(2,9,4,11),(5,13,7,15),(6,16,8,14)], [(1,5,11,14),(2,8,10,15),(3,7,9,16),(4,6,12,13)]])
 
G:=TransitiveGroup(16,263);
 

Matrix representation of C4.4D4⋊C4 ►in GL8(ℤ)

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

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],[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],[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,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0],[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,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C4.4D4⋊C4 in GAP, Magma, Sage, TeX 

C_4._4D_4\rtimes C_4
 
% in TeX
 
G:=Group("C4.4D4:C4");
 
// GroupNames label
 
G:=SmallGroup(128,860);
 
// by ID
 
G=gap.SmallGroup(128,860);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,352,1123,1018,248,1971,375,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^4=1,c^2=a^2,d*b*d^-1=a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,d*c*d^-1=a^2*c>;
 
// generators/relations
 

Export

Character table of C4.4D4⋊C4 in TeX

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