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G = C42.C2order 32 = 25

4th non-split extension by C42 of C2 acting faithfully

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

Aliases: C4.3Q8, C42.4C2, C22.15C23, C4⋊C4.4C2, C2.4(C2×Q8), C2.8(C4○D4), (C2×C4).3C22, SmallGroup(32,32)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.C2
C1C2C22C2×C4C42 — C42.C2
C1C22 — C42.C2
C1C22 — C42.C2
C1C22 — C42.C2

Generators and relations for C42.C2
 G = < a,b,c | a4=b4=1, c2=b2, ab=ba, cac-1=ab2, cbc-1=a2b >

2C4
2C4
2C4
2C4
2C4
2C4

Character table of C42.C2

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J
 size 11112222224444
ρ111111111111111    trivial
ρ21111-1-1-11-1111-1-1    linear of order 2
ρ311111-11-1-1-11-1-11    linear of order 2
ρ41111-11-1-11-11-11-1    linear of order 2
ρ51111-11-1-11-1-11-11    linear of order 2
ρ611111-11-1-1-1-111-1    linear of order 2
ρ71111-1-1-11-11-1-111    linear of order 2
ρ81111111111-1-1-1-1    linear of order 2
ρ92-2-2220-20000000    symplectic lifted from Q8, Schur index 2
ρ102-2-22-2020000000    symplectic lifted from Q8, Schur index 2
ρ1122-2-2000-2i02i0000    complex lifted from C4○D4
ρ122-22-202i00-2i00000    complex lifted from C4○D4
ρ132-22-20-2i002i00000    complex lifted from C4○D4
ρ1422-2-20002i0-2i0000    complex lifted from C4○D4

Smallest permutation representation of C42.C2
Regular action on 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 17 15 6)(2 18 16 7)(3 19 13 8)(4 20 14 5)(9 29 24 28)(10 30 21 25)(11 31 22 26)(12 32 23 27)
(1 26 15 31)(2 32 16 27)(3 28 13 29)(4 30 14 25)(5 12 20 23)(6 24 17 9)(7 10 18 21)(8 22 19 11)

G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,29,24,28)(10,30,21,25)(11,31,22,26)(12,32,23,27), (1,26,15,31)(2,32,16,27)(3,28,13,29)(4,30,14,25)(5,12,20,23)(6,24,17,9)(7,10,18,21)(8,22,19,11)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,29,24,28)(10,30,21,25)(11,31,22,26)(12,32,23,27), (1,26,15,31)(2,32,16,27)(3,28,13,29)(4,30,14,25)(5,12,20,23)(6,24,17,9)(7,10,18,21)(8,22,19,11) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,17,15,6),(2,18,16,7),(3,19,13,8),(4,20,14,5),(9,29,24,28),(10,30,21,25),(11,31,22,26),(12,32,23,27)], [(1,26,15,31),(2,32,16,27),(3,28,13,29),(4,30,14,25),(5,12,20,23),(6,24,17,9),(7,10,18,21),(8,22,19,11)])

Matrix representation of C42.C2 in GL4(𝔽5) generated by

1300
0400
0020
0003
,
3000
0300
0030
0002
,
4200
4100
0001
0040
G:=sub<GL(4,GF(5))| [1,0,0,0,3,4,0,0,0,0,2,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[4,4,0,0,2,1,0,0,0,0,0,4,0,0,1,0] >;

C42.C2 in GAP, Magma, Sage, TeX

C_4^2.C_2
% in TeX

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

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

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

G:=PCGroup([5,-2,2,2,-2,2,40,101,86,302,42]);
// Polycyclic

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

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