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G = C22×D4order 32 = 25

Direct product of C22 and D4

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

Aliases: C22×D4, C4⋊C23, C242C2, C22⋊C23, C2.1C24, C233C22, (C2×C4)⋊4C22, (C22×C4)⋊5C2, 2-Sylow(GO-(4,3)), SmallGroup(32,46)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22×D4
C1C2C22C23C24 — C22×D4
C1C2 — C22×D4
C1C23 — C22×D4
C1C2 — C22×D4

Generators and relations for C22×D4
 G = < a,b,c,d | a2=b2=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 158 in 118 conjugacy classes, 78 normal (5 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C22 [×15], C22 [×24], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C22×C4, C2×D4 [×12], C24 [×2], C22×D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4

Character table of C22×D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211-111-1-1-11-1-11-111-1-111-1    linear of order 2
ρ31-1-1-111-11-1-11111-1-1-1-111    linear of order 2
ρ41-11-11-11-1-11-11-11-111-11-1    linear of order 2
ρ511111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ61-11-11-11-11-11-11-11-11-11-1    linear of order 2
ρ711-111-1-1-1-111-11-1-11-111-1    linear of order 2
ρ81-1-1-111-1111-1-1-1-111-1-111    linear of order 2
ρ9111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ101-11-11-11-1-11-111-11-1-11-11    linear of order 2
ρ1111-111-1-1-11-1-111-1-111-1-11    linear of order 2
ρ121-1-1-111-11-1-111-1-11111-1-1    linear of order 2
ρ1311-111-1-1-1-111-1-111-11-1-11    linear of order 2
ρ141-1-1-111-1111-1-111-1-111-1-1    linear of order 2
ρ1511111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ161-11-11-11-11-11-1-11-11-11-11    linear of order 2
ρ1722-2-2-2-222000000000000    orthogonal lifted from D4
ρ18222-2-22-2-2000000000000    orthogonal lifted from D4
ρ192-222-2-2-22000000000000    orthogonal lifted from D4
ρ202-2-22-222-2000000000000    orthogonal lifted from D4

Permutation representations of C22×D4
On 16 points - transitive group 16T25
Generators in S16
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 9)(6 12)(7 11)(8 10)

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

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

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

G:=TransitiveGroup(16,25);

Matrix representation of C22×D4 in GL4(ℤ) generated by

-1000
0-100
0010
0001
,
-1000
0100
00-10
000-1
,
1000
0100
0001
00-10
,
1000
0-100
00-10
0001
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[-1,0,0,0,0,1,0,0,0,0,-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] >;

C22×D4 in GAP, Magma, Sage, TeX

C_2^2\times D_4
% in TeX

G:=Group("C2^2xD4");
// GroupNames label

G:=SmallGroup(32,46);
// by ID

G=gap.SmallGroup(32,46);
# by ID

G:=PCGroup([5,-2,2,2,2,-2,181]);
// Polycyclic

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

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