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## G = C4⋊1D4order 32 = 25

### The semidirect product of C4 and D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C41D4, C426C2, C23.5C22, C22.17C23, (C2×D4)⋊3C2, C2.9(C2×D4), (C2×C4).23C22, SmallGroup(32,34)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C41D4
G = < a,b,c | a4=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 90 in 54 conjugacy classes, 26 normal (4 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C42, C2×D4, C41D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C41D4

Character table of C41D4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F size 1 1 1 1 4 4 4 4 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 -2 2 -2 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 2 0 0 0 -2 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 0 0 0 0 0 -2 0 0 0 2 orthogonal lifted from D4 ρ13 2 2 -2 -2 0 0 0 0 0 2 0 0 0 -2 orthogonal lifted from D4 ρ14 2 -2 -2 2 0 0 0 0 -2 0 0 0 2 0 orthogonal lifted from D4

Permutation representations of C41D4
On 16 points - transitive group 16T51
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 14 11)(2 7 15 12)(3 8 16 9)(4 5 13 10)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)

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

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

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

G:=TransitiveGroup(16,51);

C41D4 is a maximal subgroup of
C4.D8  C42⋊C4  D44D4  C4⋊D8  C4⋊SD16  C4.4D8  C42.29C22  C22.26C24  C22.29C24  C22.34C24  D42  Q86D4  C22.53C24  C22.54C24  C23.A4  C4⋊S3≀C2
C4p⋊D4: C85D4  C84D4  C83D4  C4⋊D12  C123D4  C204D4  C20⋊D4  C284D4 ...
C41D4 is a maximal quotient of
C429C4  C24.3C22  C232D4  C23.4Q8  C4⋊Q16  C8.12D4  C8.2D4  C4⋊S3≀C2
C4p⋊D4: C85D4  C84D4  C83D4  C4⋊D12  C123D4  C204D4  C20⋊D4  C284D4 ...

Matrix representation of C41D4 in GL4(ℤ) generated by

 0 1 0 0 -1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 0
,
 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 1
G:=sub<GL(4,Integers())| [0,-1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0],[1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1] >;

C41D4 in GAP, Magma, Sage, TeX

C_4\rtimes_1D_4
% in TeX

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

G:=SmallGroup(32,34);
// by ID

G=gap.SmallGroup(32,34);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,101,46,302,72]);
// Polycyclic

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

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