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G = C2×C42⋊C4order 128 = 27

Direct product of C2 and C42⋊C4

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

Aliases: C2×C42⋊C4, C24.37D4, C424(C2×C4), (C2×C42)⋊9C4, C41D418C4, C23.7(C2×D4), (C2×D4).129D4, (C22×D4)⋊11C4, C23⋊C43C22, (C2×D4).18C23, C41D4.133C22, 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×C41D4).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

C1C2×C4 — C2×C42⋊C4
C1C2C22C23C2×D4C22×D4C2×C41D4 — C2×C42⋊C4
C1C2C22C2×C4 — C2×C42⋊C4
C1C22C23C22×D4 — C2×C42⋊C4
C1C2C22C2×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 [×2], C2 [×8], C4 [×10], C22 [×3], C22 [×26], C2×C4 [×2], C2×C4 [×16], D4 [×24], C23, C23 [×4], C23 [×16], C42 [×2], C42, C22⋊C4 [×6], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×21], C24 [×2], C24, C23⋊C4 [×4], C23⋊C4 [×2], C2×C42, C2×C22⋊C4 [×2], C41D4 [×4], C41D4 [×2], C22×D4 [×2], C22×D4 [×2], C42⋊C4 [×4], C2×C23⋊C4 [×2], C2×C41D4, C2×C42⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C42⋊C4 [×2], 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 5 7 4)(2 6 8 3)(9 13 11 15)(10 14 12 16)
(1 12 3 11)(2 15 4 14)(5 16 8 13)(6 9 7 10)

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

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

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

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

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

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

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

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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