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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 8)(2 10 16 5)(3 11 13 6)(4 12 14 7)
(1 3)(2 14)(4 16)(5 12)(6 8)(7 10)(9 11)(13 15)

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

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

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

G:=TransitiveGroup(16,17);

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

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