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G = C42⋊C2order 32 = 25

1st semidirect product of C42 and C2 acting faithfully

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

Aliases: C421C2, C22.6C23, C23.7C22, C4(C4⋊C4), C4⋊C46C2, (C2×C4)⋊4C4, C4.9(C2×C4), C4(C22⋊C4), C2.1(C4○D4), C22⋊C4.3C2, C22.5(C2×C4), C2.3(C22×C4), (C22×C4).4C2, (C2×C4).10C22, SmallGroup(32,24)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C42⋊C2
C1C2C22C2×C4C22×C4 — C42⋊C2
C1C2 — C42⋊C2
C1C2×C4 — C42⋊C2
C1C22 — C42⋊C2

Generators and relations for C42⋊C2
 G = < a,b,c | a4=b4=c2=1, ab=ba, cac=ab2, bc=cb >

2C2
2C2
2C4
2C22
2C4
2C22
2C4
2C4

Character table of C42⋊C2

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11112211112222222222
ρ111111111111111111111    trivial
ρ21111-1-111111-11-1-1-111-1-1    linear of order 2
ρ3111111-1-1-1-1-111-1-1-1-11-11    linear of order 2
ρ41111-1-1-1-1-1-1-1-11111-111-1    linear of order 2
ρ51111-1-11111-11-11-1-1-1-111    linear of order 2
ρ6111111-1-1-1-11-1-11-1-11-11-1    linear of order 2
ρ71111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ81111-1-1-1-1-1-111-1-1111-1-11    linear of order 2
ρ91-11-1-1111-1-1-iii-i-11i-ii-i    linear of order 4
ρ101-11-1-11-1-111iiii1-1-i-i-i-i    linear of order 4
ρ111-11-11-111-1-1ii-i-i1-1-iii-i    linear of order 4
ρ121-11-11-1-1-111-ii-ii-11ii-i-i    linear of order 4
ρ131-11-11-1-1-111i-ii-i-11-i-iii    linear of order 4
ρ141-11-11-111-1-1-i-iii1-1i-i-ii    linear of order 4
ρ151-11-1-11-1-111-i-i-i-i1-1iiii    linear of order 4
ρ161-11-1-1111-1-1i-i-ii-11-ii-ii    linear of order 4
ρ1722-2-2002i-2i-2i2i0000000000    complex lifted from C4○D4
ρ1822-2-200-2i2i2i-2i0000000000    complex lifted from C4○D4
ρ192-2-22002i-2i2i-2i0000000000    complex lifted from C4○D4
ρ202-2-2200-2i2i-2i2i0000000000    complex lifted from C4○D4

Permutation representations of C42⋊C2
On 16 points - transitive group 16T17
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 15 6)(2 10 16 7)(3 11 13 8)(4 12 14 5)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)

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

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

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

G:=TransitiveGroup(16,17);

C42⋊C2 is a maximal subgroup of
C4×C4○D4  C22.11C24  C23.32C23  C23.33C23  C22.19C24  C23.36C23  C23.37C23  C22.29C24  C22.34C24  C22.35C24  C22.36C24  C23.41C23  C22.45C24  C22.46C24  C22.47C24  C22.49C24  C22.50C24  D10.C23  (C6×C12)⋊5C4  D26.C23
 C23.D2p: C4.9C42  C426C4  C23.24D4  C23.37D4  C23.38D4  C42⋊C22  C23.25D4  M4(2)⋊C4 ...
 (C4×C4p)⋊C2: C82M4(2)  C42.7C22  C422S3  C42⋊D5  C42⋊D7  C42⋊D11  C42⋊D13 ...
 (C2×C4).D2p: M4(2)⋊4C4  C23.C23  C42.6C22  C23.38C23  C4⋊C47S3  C4⋊C47D5  C4⋊C47D7  C4⋊C47D11 ...
C42⋊C2 is a maximal quotient of
C424C4  C425C4  C42.6C4  D10.C23  (C6×C12)⋊5C4  D26.C23
 C23.D2p: C23.7Q8  C23.34D4  C23.16D6  C23.26D6  C23.11D10  C23.21D10  C23.11D14  C23.21D14 ...
 (C4×C4p)⋊C2: C42.12C4  C42.7C22  C422S3  C42⋊D5  C42⋊D7  C42⋊D11  C42⋊D13 ...
 (C2×C4).D2p: C4×C22⋊C4  C4×C4⋊C4  C428C4  C23.63C23  C24.C22  C4⋊C47S3  C4⋊C47D5  C4⋊C47D7 ...

Matrix representation of C42⋊C2 in GL3(𝔽5) generated by

200
023
003
,
400
020
002
,
400
010
024
G:=sub<GL(3,GF(5))| [2,0,0,0,2,0,0,3,3],[4,0,0,0,2,0,0,0,2],[4,0,0,0,1,2,0,0,4] >;

C42⋊C2 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_2
% in TeX

G:=Group("C4^2:C2");
// GroupNames label

G:=SmallGroup(32,24);
// by ID

G=gap.SmallGroup(32,24);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101,42]);
// Polycyclic

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

Export

Subgroup lattice of C42⋊C2 in TeX
Character table of C42⋊C2 in TeX

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