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G = C42.32Q8  order 128 = 27

32nd non-split extension by C42 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.32Q8, (C2×C8).9D4, C4.9C42.2C2, C42⋊6C4.14C2, C4⋊M4(2).1C2, C4.28(C42.C2), M4(2).C4.4C2, C4.103(C4.4D4), C23.136(C4○D4), M4(2)⋊4C4.6C2, (C22×C4).744C23, (C2×C42).399C22, C22.40(C22⋊Q8), C42⋊C2.74C22, C22.12(C42⋊2C2), C4.122(C22.D4), (C2×M4(2)).240C22, C2.8(C23.83C23), (C2×C4).119(C2×Q8), (C2×C4).1386(C2×D4), (C2×C4).787(C4○D4), SmallGroup(128,834)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22×C4 — C42.32Q8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — M4(2)⋊4C4 — C42.32Q8
Lower central
C1 — C2 — C22×C4 — C42.32Q8
Upper central
C1 — C4 — C22×C4 — C42.32Q8
Jennings
C1 — C2 — C2 — C22×C4 — C42.32Q8

Generators and relations for C42.32Q8
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2c2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=dbd-1=b-1, dcd-1=b2c3 >

Subgroups: 152 in 79 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C4⋊C8, C8.C4, C2×C42, C42⋊C2, C2×M4(2), C2×M4(2), C4.9C42, C42⋊6C4, M4(2)⋊4C4, C4⋊M4(2), M4(2).C4, C42.32Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C23.83C23, C42.32Q8

Character table of C42.32Q8

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F8G8H
 size 11222112224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-1-1-11111-111-1    linear of order 2
ρ311111111111111-1-1-1-111-1-11-1-11    linear of order 2
ρ41111111111-1-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-1-111-1-1-1-1111-11    linear of order 2
ρ6111111111111111-1-11-1-1-11-11-1-1    linear of order 2
ρ71111111111-1-1-1-11-1-11-1-11-11-111    linear of order 2
ρ811111111111111-111-1-1-11-1-1-11-1    linear of order 2
ρ9222-2-222-22-200000000-22000000    orthogonal lifted from D4
ρ10222-2-222-22-2000000002-2000000    orthogonal lifted from D4
ρ11222-2-2-2-22-22-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ12222-2-2-2-22-2222-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1322-22-2-2-222-20000000000-2i0002i0    complex lifted from C4○D4
ρ1422222-2-2-2-2-20000000000002i00-2i    complex lifted from C4○D4
ρ1522-2-22222-2-2000002i-2i000000000    complex lifted from C4○D4
ρ1622-22-222-2-2200002i00-2i00000000    complex lifted from C4○D4
ρ1722222-2-2-2-2-2000000000000-2i002i    complex lifted from C4○D4
ρ1822-2-22-2-2-22200000000000-2i02i00    complex lifted from C4○D4
ρ1922-2-22-2-2-222000000000002i0-2i00    complex lifted from C4○D4
ρ2022-22-2-2-222-200000000002i000-2i0    complex lifted from C4○D4
ρ2122-22-222-2-220000-2i002i00000000    complex lifted from C4○D4
ρ2222-2-22222-2-200000-2i2i000000000    complex lifted from C4○D4
ρ234-4000-4i4i000-22-2i2i000000000000    complex faithful
ρ244-40004i-4i0002-2-2i2i000000000000    complex faithful
ρ254-4000-4i4i0002-22i-2i000000000000    complex faithful
ρ264-40004i-4i000-222i-2i000000000000    complex faithful

Permutation representations of C42.32Q8
►On 16 points - transitive group 16T334
Generators in S16
(9 11 13 15)(10 16 14 12)

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

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

(1 13 7 15 5 9 3 11)(2 12 8 14 6 16 4 10)
 
G:=sub<Sym(16)| (9,11,13,15)(10,16,14,12), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,7,15,5,9,3,11)(2,12,8,14,6,16,4,10)>;
 
G:=Group( (9,11,13,15)(10,16,14,12), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,7,15,5,9,3,11)(2,12,8,14,6,16,4,10) );
 
G=PermutationGroup([[(9,11,13,15),(10,16,14,12)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13,7,15,5,9,3,11),(2,12,8,14,6,16,4,10)]])
 
G:=TransitiveGroup(16,334);
 

Matrix representation of C42.32Q8 ►in GL4(𝔽5) generated by

4004
0010
0240
1004
,
2000
0300
0030
0002
,
0220
2002
3000
0020
,
0430
0002
1004
0400
G:=sub<GL(4,GF(5))| [4,0,0,1,0,0,2,0,0,1,4,0,4,0,0,4],[2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[0,2,3,0,2,0,0,0,2,0,0,2,0,2,0,0],[0,0,1,0,4,0,0,4,3,0,0,0,0,2,4,0] >;
 

C42.32Q8 in GAP, Magma, Sage, TeX 

C_4^2._{32}Q_8
 
% in TeX
 
G:=Group("C4^2.32Q8");
 
// GroupNames label
 
G:=SmallGroup(128,834);
 
// by ID
 
G=gap.SmallGroup(128,834);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,58,1018,248,1411,4037,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2*c^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^3>;
 
// generators/relations
 

Export

Character table of C42.32Q8 in TeX

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