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G = C22.D4order 32 = 25

3rd non-split extension by C22 of D4 acting via D4/C22=C2

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

Aliases: C22.4D4, C22.13C23, C23.10C22, C4⋊C44C2, C2.7(C2×D4), C22⋊C44C2, (C22×C4)⋊3C2, (C2×D4).4C2, C2.6(C4○D4), (C2×C4).13C22, SmallGroup(32,30)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.D4
C1C2C22C2×C4C22×C4 — C22.D4
C1C22 — C22.D4
C1C22 — C22.D4
C1C22 — C22.D4

Generators and relations for C22.D4
 G = < a,b,c,d | a2=b2=c4=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=bc-1 >

2C2
2C2
4C2
2C22
2C4
2C22
2C22
2C22
2C22
2C4
2C4
2C4
2C4
2D4
2D4
2C2×C4
2C2×C4

Character table of C22.D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G
 size 11112242222444
ρ111111111111111    trivial
ρ21111111-1-1-1-1-11-1    linear of order 2
ρ3111111-11111-1-1-1    linear of order 2
ρ4111111-1-1-1-1-11-11    linear of order 2
ρ51111-1-11-111-11-1-1    linear of order 2
ρ61111-1-111-1-11-1-11    linear of order 2
ρ71111-1-1-1-111-1-111    linear of order 2
ρ81111-1-1-11-1-1111-1    linear of order 2
ρ92-22-22-200000000    orthogonal lifted from D4
ρ102-22-2-2200000000    orthogonal lifted from D4
ρ1122-2-200002i-2i0000    complex lifted from C4○D4
ρ1222-2-20000-2i2i0000    complex lifted from C4○D4
ρ132-2-220002i00-2i000    complex lifted from C4○D4
ρ142-2-22000-2i002i000    complex lifted from C4○D4

Permutation representations of C22.D4
On 16 points - transitive group 16T37
Generators in S16
(1 16)(2 7)(3 14)(4 5)(6 10)(8 12)(9 15)(11 13)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 9)(4 11)(5 7)(6 16)(8 14)(13 15)

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

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

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

G:=TransitiveGroup(16,37);

On 16 points - transitive group 16T54
Generators in S16
(2 14)(4 16)(5 12)(7 10)
(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 10)(2 6)(3 12)(4 8)(5 15)(7 13)(9 14)(11 16)

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

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

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

G:=TransitiveGroup(16,54);

C22.D4 is a maximal subgroup of
C23.36C23  C233D4  C22.32C24  C22.33C24  C22.34C24  C22.36C24  C22.45C24  C22.46C24  C22.47C24  C22.53C24  C22.54C24  C22.56C24  C22.57C24  C62.9D4
 C23.D2p: C23.D4  C23.7D4  C23.9D6  C23.21D6  C23.28D6  C23.23D6  D10.12D4  C22.D20 ...
 C2p.(C2×D4): C22.19C24  C23.38C23  D45D4  D46D4  D6.D4  D10.13D4  D14.5D4  D22.5D4 ...
C22.D4 is a maximal quotient of
C62.9D4
 C23.D2p: C23.34D4  C23.8Q8  C23.23D4  C23.10D4  C23.11D4  C22.D8  C23.46D4  C23.19D4 ...
 (C2×C4).D2p: C23.63C23  C24.C22  C23.81C23  C23.4Q8  C23.83C23  D6.D4  D10.13D4  D14.5D4 ...

Matrix representation of C22.D4 in GL4(𝔽5) generated by

1000
0100
0002
0030
,
1000
0100
0040
0004
,
1200
4400
0004
0040
,
1000
4400
0010
0004
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,0,3,0,0,2,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,4,0,0,2,4,0,0,0,0,0,4,0,0,4,0],[1,4,0,0,0,4,0,0,0,0,1,0,0,0,0,4] >;

C22.D4 in GAP, Magma, Sage, TeX

C_2^2.D_4
% in TeX

G:=Group("C2^2.D4");
// GroupNames label

G:=SmallGroup(32,30);
// by ID

G=gap.SmallGroup(32,30);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,101,302,42]);
// Polycyclic

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

Export

Subgroup lattice of C22.D4 in TeX
Character table of C22.D4 in TeX

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