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G = C422C2order 32 = 25

2nd semidirect product of C42 and C2 acting faithfully

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

Aliases: C422C2, C23.4C22, C22.16C23, C4⋊C45C2, C2.9(C4○D4), C22⋊C4.2C2, (C2×C4).4C22, SmallGroup(32,33)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C422C2
C1C2C22C2×C4C42 — C422C2
C1C22 — C422C2
C1C22 — C422C2
C1C22 — C422C2

Generators and relations for C422C2
 G = < a,b,c | a4=b4=c2=1, ab=ba, cac=ab2, cbc=a2b-1 >

4C2
2C22
2C22
2C4
2C22
2C4
2C4
2C4
2C4
2C4

Character table of C422C2

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I
 size 11114222222444
ρ111111111111111    trivial
ρ211111-1-1-1-111-11-1    linear of order 2
ρ31111-1-1-1-1-1111-11    linear of order 2
ρ41111-1111111-1-1-1    linear of order 2
ρ51111111-1-1-1-1-1-11    linear of order 2
ρ611111-1-111-1-11-1-1    linear of order 2
ρ71111-1-1-111-1-1-111    linear of order 2
ρ81111-111-1-1-1-111-1    linear of order 2
ρ92-22-200000-2i2i000    complex lifted from C4○D4
ρ102-2-2202i-2i0000000    complex lifted from C4○D4
ρ1122-2-2000-2i2i00000    complex lifted from C4○D4
ρ122-22-2000002i-2i000    complex lifted from C4○D4
ρ132-2-220-2i2i0000000    complex lifted from C4○D4
ρ1422-2-20002i-2i00000    complex lifted from C4○D4

Permutation representations of C422C2
On 16 points - transitive group 16T27
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 5 11)(2 14 6 12)(3 15 7 9)(4 16 8 10)
(2 6)(4 8)(9 13)(10 12)(11 15)(14 16)

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

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

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

G:=TransitiveGroup(16,27);

Matrix representation of C422C2 in GL4(𝔽5) generated by

0100
4000
0002
0030
,
2000
0200
0001
0040
,
1000
0400
0010
0004
G:=sub<GL(4,GF(5))| [0,4,0,0,1,0,0,0,0,0,0,3,0,0,2,0],[2,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4] >;

C422C2 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2C_2
% in TeX

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

G:=SmallGroup(32,33);
// by ID

G=gap.SmallGroup(32,33);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,101,126,302,42]);
// Polycyclic

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

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