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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

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 [×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 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N size 1 1 1 1 2 2 4 4 4 4 8 8 4 4 4 4 4 4 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ6 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ7 1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ8 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 i i -i -i -i -i i i linear of order 4 ρ10 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 i i -i -i i i -i -i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -i -i i i i i -i -i linear of order 4 ρ12 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -i -i i i -i -i i i linear of order 4 ρ13 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 i -i -i i -i i i -i linear of order 4 ρ14 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 i -i -i i i -i -i i linear of order 4 ρ15 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -i i i -i i -i -i i linear of order 4 ρ16 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -i i i -i -i i i -i linear of order 4 ρ17 2 2 2 2 2 2 -2 2 2 -2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 2 -2 2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 2 -2 -2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 0 0 0 0 0 0 -2 2 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ22 4 4 -4 -4 0 0 0 0 0 0 0 0 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ23 4 -4 -4 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 -2 2 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C42⋊C4 ρ25 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ26 4 -4 4 -4 0 0 0 0 0 0 0 0 2 -2 0 0 2 -2 0 0 0 0 0 0 0 0 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)

 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 0 0 0 0 0 1 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 4 0
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0
,
 3 4 0 0 0 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 4 0 0 0

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

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