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G = C41D4order 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

C1C22 — C41D4
C1C2C22C2×C4C42 — C41D4
C1C22 — C41D4
C1C22 — C41D4
C1C22 — C41D4

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 [×3], C2 [×4], C4 [×6], C22, C22 [×12], C2×C4 [×3], D4 [×12], C23 [×4], C42, C2×D4 [×6], C41D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4

Character table of C41D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F
 size 11114444222222
ρ111111111111111    trivial
ρ21111-111-1-11-1-1-11    linear of order 2
ρ311111-11-1-1-111-1-1    linear of order 2
ρ41111-1-1111-1-1-11-1    linear of order 2
ρ51111-11-11-1-111-1-1    linear of order 2
ρ6111111-1-11-1-1-11-1    linear of order 2
ρ71111-1-1-1-1111111    linear of order 2
ρ811111-1-11-11-1-1-11    linear of order 2
ρ92-22-20000002-200    orthogonal lifted from D4
ρ102-2-2200002000-20    orthogonal lifted from D4
ρ112-22-2000000-2200    orthogonal lifted from D4
ρ1222-2-200000-20002    orthogonal lifted from D4
ρ1322-2-2000002000-2    orthogonal lifted from D4
ρ142-2-220000-200020    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);

Matrix representation of C41D4 in GL4(ℤ) generated by

0100
-1000
0010
0001
,
1000
0100
0001
00-10
,
1000
0-100
00-10
0001
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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