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## G = C2×C2≀C4order 128 = 27

### Direct product of C2 and C2≀C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C2≀C4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — C22×D4 — C2×C22≀C2 — C2×C2≀C4
 Lower central C1 — C2 — C22 — C23 — C2×C2≀C4
 Upper central C1 — C22 — C23 — C22×D4 — C2×C2≀C4
 Jennings C1 — C2 — C22 — C2×D4 — C2×C2≀C4

Generators and relations for C2×C2≀C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 660 in 219 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×6], C22 [×3], C22 [×49], C8 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×12], C23 [×5], C23 [×46], C22⋊C4 [×2], C22⋊C4 [×8], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C24 [×2], C24 [×2], C24 [×8], C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.D4, C2×C22⋊C4, C2×C22⋊C4 [×2], C22≀C2 [×4], C22≀C2 [×2], C2×M4(2), C22×D4, C22×D4, C25, C2≀C4 [×4], C2×C23⋊C4, C2×C4.D4, C2×C22≀C2, C2×C2≀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, C2≀C4 [×2], C2×C23⋊C4, C2×C2≀C4

Character table of C2×C2≀C4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 8 8 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 i -i 1 i 1 -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 -i -i -1 i 1 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 i i -1 -i 1 -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 -i i 1 -i 1 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 -i -i 1 i -1 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 i -i -1 i -1 -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 -i i -1 -i -1 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 i i 1 -i -1 -i i -i i -i linear of order 4 ρ17 2 -2 2 -2 2 -2 0 2 0 0 -2 -2 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 -2 2 -2 0 2 0 0 2 -2 0 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 0 -2 0 0 2 -2 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 0 -2 0 0 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 -4 4 0 0 -2 0 2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ22 4 4 -4 -4 0 0 -2 0 -2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀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 -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 ρ25 4 -4 -4 4 0 0 2 0 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ26 4 4 -4 -4 0 0 2 0 2 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4

Permutation representations of C2×C2≀C4
On 16 points - transitive group 16T227
Generators in S16
(1 15)(2 16)(3 13)(4 14)(5 7)(6 8)(9 11)(10 12)
(1 3)(2 7)(4 11)(5 16)(6 10)(8 12)(9 14)(13 15)
(1 8)(2 11)(3 12)(4 7)(5 14)(6 15)(9 16)(10 13)
(2 14)(4 16)(5 11)(7 9)
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,227);

On 16 points - transitive group 16T259
Generators in S16
(1 10)(2 11)(3 12)(4 9)(5 16)(6 13)(7 14)(8 15)
(1 10)(4 7)(8 15)(9 14)
(1 15)(2 11)(4 7)(5 16)(8 10)(9 14)
(1 10)(2 5)(3 12)(4 7)(6 13)(8 15)(9 14)(11 16)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,259);

On 16 points - transitive group 16T261
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(2 15)(7 12)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,261);

On 16 points - transitive group 16T273
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)
(1 5)(2 16)(3 7)(6 12)(9 13)(11 15)
(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,273);

On 16 points - transitive group 16T283
Generators in S16
(1 7)(2 8)(3 6)(4 5)(9 14)(10 15)(11 16)(12 13)
(1 6)(2 12)(3 7)(4 15)(5 10)(8 13)(9 11)(14 16)
(1 11)(2 10)(3 14)(4 13)(5 12)(6 9)(7 16)(8 15)
(2 5)(4 8)(10 12)(13 15)
(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,283);

Matrix representation of C2×C2≀C4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1
,
 1 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 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 16 0 0 0 0 16 0 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,16,0,0] >;

C2×C2≀C4 in GAP, Magma, Sage, TeX

C_2\times C_2\wr C_4
% in TeX

G:=Group("C2xC2wrC4");
// GroupNames label

G:=SmallGroup(128,850);
// by ID

G=gap.SmallGroup(128,850);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,851,375,4037]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations

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