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G = C2×C42⋊C4  order 128 = 27

Direct product of C2 and C42⋊C4

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

Aliases: C2×C42⋊C4, C24.37D4, C42⋊4(C2×C4), (C2×C42)⋊9C4, C4⋊1D4⋊18C4, C23.7(C2×D4), (C2×D4).129D4, (C22×D4)⋊11C4, C23⋊C4⋊3C22, (C2×D4).18C23, C4⋊1D4.133C22, C22.50(C23⋊C4), C23.22(C22⋊C4), (C22×D4).101C22, (C2×D4)⋊3(C2×C4), (C2×C23⋊C4)⋊13C2, C2.36(C2×C23⋊C4), (C2×C4⋊1D4).11C2, (C2×C4).93(C22×C4), (C22×C4).79(C2×C4), (C2×C4).49(C22⋊C4), C22.60(C2×C22⋊C4), SmallGroup(128,856)

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

Derived series
C1 — C2×C4 — C2×C42⋊C4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C22×D4 — C2×C4⋊1D4 — C2×C42⋊C4
Lower central
C1 — C2 — C22 — C2×C4 — C2×C42⋊C4
Upper central
C1 — C22 — C23 — C22×D4 — C2×C42⋊C4
Jennings
C1 — C2 — C22 — C2×D4 — C2×C42⋊C4

Generators and relations for C2×C42⋊C4
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 564 in 184 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, C23⋊C4, C23⋊C4, C2×C42, C2×C22⋊C4, C4⋊1D4, C4⋊1D4, C22×D4, C22×D4, C42⋊C4, C2×C23⋊C4, C2×C4⋊1D4, C2×C42⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C42⋊C4, C2×C23⋊C4, C2×C42⋊C4

Character table of C2×C42⋊C4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11112244448844444488888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-111-1-11111-1-1-1-1    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-111-1-1-1-1-1-11111    linear of order 2
ρ51-1-11-11-1-111-11-1-1-11111-11-11-11-1    linear of order 2
ρ61-1-11-11-1-1111-111-11-1-11-11-1-11-11    linear of order 2
ρ71-1-11-11-1-111-11-1-1-1111-11-11-11-11    linear of order 2
ρ81-1-11-11-1-1111-111-11-1-1-11-111-11-1    linear of order 2
ρ9111111-1-1-1-111-1-111-1-1ii-i-i-i-iii    linear of order 4
ρ10111111-1-1-1-1-1-1111111ii-i-iii-i-i    linear of order 4
ρ11111111-1-1-1-111-1-111-1-1-i-iiiii-i-i    linear of order 4
ρ12111111-1-1-1-1-1-1111111-i-iii-i-iii    linear of order 4
ρ131-1-11-1111-1-1-1111-11-1-1i-i-ii-iii-i    linear of order 4
ρ141-1-11-1111-1-11-1-1-1-1111i-i-iii-i-ii    linear of order 4
ρ151-1-11-1111-1-1-1111-11-1-1-iii-ii-i-ii    linear of order 4
ρ161-1-11-1111-1-11-1-1-1-1111-iii-i-iii-i    linear of order 4
ρ17222222-222-20000-2-20000000000    orthogonal lifted from D4
ρ182-2-22-22-22-2200002-20000000000    orthogonal lifted from D4
ρ192-2-22-222-22-200002-20000000000    orthogonal lifted from D4
ρ202222222-2-220000-2-20000000000    orthogonal lifted from D4
ρ214-44-400000000-2200-2200000000    orthogonal lifted from C42⋊C4
ρ2244-4-4000000002-200-2200000000    orthogonal lifted from C42⋊C4
ρ234-4-444-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2444-4-400000000-22002-200000000    orthogonal lifted from C42⋊C4
ρ254444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ264-44-4000000002-2002-200000000    orthogonal lifted from C42⋊C4

Permutation representations of C2×C42⋊C4
►On 16 points - transitive group 16T235
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 16)(10 13)(11 14)(12 15)

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

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

(1 10 8 9)(2 13 7 16)(3 14 6 15)(4 11 5 12)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,16)(10,13)(11,14)(12,15), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,5,7)(2,4,6,8)(9,13,11,15)(10,14,12,16), (1,10,8,9)(2,13,7,16)(3,14,6,15)(4,11,5,12)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,16)(10,13)(11,14)(12,15), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,5,7)(2,4,6,8)(9,13,11,15)(10,14,12,16), (1,10,8,9)(2,13,7,16)(3,14,6,15)(4,11,5,12) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,16),(10,13),(11,14),(12,15)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,5,7),(2,4,6,8),(9,13,11,15),(10,14,12,16)], [(1,10,8,9),(2,13,7,16),(3,14,6,15),(4,11,5,12)]])
 
G:=TransitiveGroup(16,235);
 

►On 16 points - transitive group 16T248
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 14)(10 15)(11 16)(12 13)

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

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

(1 16 8 12)(2 11 7 13)(3 15 5 9)(4 10 6 14)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,14)(10,15)(11,16)(12,13), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,6,4)(2,7,5,3)(9,15,11,13)(10,16,12,14), (1,16,8,12)(2,11,7,13)(3,15,5,9)(4,10,6,14)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,14)(10,15)(11,16)(12,13), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,6,4)(2,7,5,3)(9,15,11,13)(10,16,12,14), (1,16,8,12)(2,11,7,13)(3,15,5,9)(4,10,6,14) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,14),(10,15),(11,16),(12,13)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,8,6,4),(2,7,5,3),(9,15,11,13),(10,16,12,14)], [(1,16,8,12),(2,11,7,13),(3,15,5,9),(4,10,6,14)]])
 
G:=TransitiveGroup(16,248);
 

Matrix representation of C2×C42⋊C4 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
140000
004000
000400
000001
000040
,
400000
040000
000100
004000
000004
000010
,
340000
020000
000040
000004
000400
004000

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,4,0,0] >;
 

C2×C42⋊C4 in GAP, Magma, Sage, TeX 

C_2\times C_4^2\rtimes C_4
 
% in TeX
 
G:=Group("C2xC4^2:C4");
 
// GroupNames label
 
G:=SmallGroup(128,856);
 
// by ID
 
G=gap.SmallGroup(128,856);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1018,248,1971,375,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^2*c>;
 
// generators/relations
 

Export

Character table of C2×C42⋊C4 in TeX

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