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## G = C4⋊1D4.C4order 128 = 27

### 5th non-split extension by C4⋊1D4 of C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4⋊1D4.C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C2×C4○D4 — C22.29C24 — C4⋊1D4.C4
 Lower central C1 — C2 — C22 — C2×C4 — C4⋊1D4.C4
 Upper central C1 — C2 — C2×C4 — C2×C4○D4 — C4⋊1D4.C4
 Jennings C1 — C2 — C22 — C2×Q8 — C4⋊1D4.C4

Generators and relations for C41D4.C4
G = < a,b,c,d | a4=b4=c2=1, d4=a2b2, ab=ba, cac=a-1, dad-1=cbc=b-1, dbd-1=a-1b2, dcd-1=a-1bc >

Subgroups: 364 in 131 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×5], C22, C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8, C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×5], C4⋊C4, C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×4], C2×D4 [×7], C2×Q8, C4○D4 [×2], C24, C4.D4 [×2], C4.10D4 [×4], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C2×M4(2) [×2], C22×D4, C2×C4○D4, C42.C4 [×4], M4(2).8C22 [×2], C22.29C24, C41D4.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×C23⋊C4, C41D4.C4

Character table of C41D4.C4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 4 4 4 8 8 2 2 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 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 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 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 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 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 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 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 linear of order 2 ρ9 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 -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 -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 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 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 -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 -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 i i i -i -i i -i -i linear of order 4 ρ17 2 2 2 -2 2 2 0 0 -2 -2 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 0 2 2 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 0 2 2 -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 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C41D4.C4
On 16 points - transitive group 16T226
Generators in S16
(1 11 5 15)(3 13 7 9)(4 8)(10 14)
(2 16 6 12)(3 7)(4 10 8 14)(9 13)
(1 11)(2 6)(3 9)(4 8)(5 15)(7 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,226);

On 16 points - transitive group 16T285
Generators in S16
(1 9 5 13)(2 4)(3 11 7 15)(6 8)(10 12)(14 16)
(1 3)(2 14 6 10)(4 16 8 12)(5 7)(9 11)(13 15)
(1 11)(2 6)(3 9)(4 8)(5 15)(7 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,285);

Matrix representation of C41D4.C4 in GL8(ℤ)

 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 0 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 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 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 0 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 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 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

G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-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,-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,-1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C41D4.C4 in GAP, Magma, Sage, TeX

C_4\rtimes_1D_4.C_4
% in TeX

G:=Group("C4:1D4.C4");
// GroupNames label

G:=SmallGroup(128,866);
// by ID

G=gap.SmallGroup(128,866);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,352,1123,1018,248,1971,375,172,4037]);
// Polycyclic

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

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