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G = C4oD4:A4  order 192 = 26·3

1st semidirect product of C4oD4 and A4 acting via A4/C22=C3

non-abelian, soluble

Aliases: C4oD4:1A4, C22:(C4.A4), Q8.1(C2xA4), Q8:A4:3C2, (C22xC4).9A4, C4.1(C22:A4), C23.23(C2xA4), (C22xQ8).7C6, (C22xC4oD4):3C3, C2.3(C2xC22:A4), SmallGroup(192,1507)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C22xQ8 — C4oD4:A4
Chief series
C1 — C2 — Q8 — C22xQ8 — Q8:A4 — C4oD4:A4
Lower central
C22xQ8 — C4oD4:A4
Upper central
C1 — C4

Generators and relations for C4oD4:A4
 G = < a,b,c,d,e,f | a4=c2=d2=e2=f3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=a2b, bd=db, be=eb, fbf-1=a-1bc, cd=dc, ce=ec, fcf-1=a-1b, fdf-1=de=ed, fef-1=d >

Subgroups: 599 in 175 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2xC4, D4, Q8, Q8, C23, C23, C12, A4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, SL2(F3), C2xA4, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C4xA4, C4.A4, C22xC4oD4, Q8:A4, C4oD4:A4
Quotients: C1, C2, C3, C6, A4, C2xA4, C4.A4, C22:A4, C2xC22:A4, C4oD4:A4

Character table of C4oD4:A4

 class 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H6A6B12A12B12C12D
 size 11336666161611336666161616161616
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ311111111ζ3ζ3211111111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ41111-1-1-1-1ζ32ζ3-1-1-1-11111ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ51111-1-1-1-1ζ3ζ32-1-1-1-11111ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ611111111ζ32ζ311111111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ72-22-20000-1-1-2i2i2i-2i000011-i-iii    complex lifted from C4.A4
ρ82-22-20000-1-12i-2i-2i2i000011ii-i-i    complex lifted from C4.A4
ρ92-22-20000ζ6ζ65-2i2i2i-2i0000ζ32ζ3ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    complex lifted from C4.A4
ρ102-22-20000ζ65ζ6-2i2i2i-2i0000ζ3ζ32ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    complex lifted from C4.A4
ρ112-22-20000ζ65ζ62i-2i-2i2i0000ζ3ζ32ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    complex lifted from C4.A4
ρ122-22-20000ζ6ζ652i-2i-2i2i0000ζ32ζ3ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    complex lifted from C4.A4
ρ1333-1-111-3100-3-311-1-1-13000000    orthogonal lifted from C2xA4
ρ143333-1-1-1-1003333-1-1-1-1000000    orthogonal lifted from A4
ρ1533-1-1-13-1-10033-1-1-1-13-1000000    orthogonal lifted from A4
ρ1633-1-1-311100-3-311-13-1-1000000    orthogonal lifted from C2xA4
ρ1733-1-1111-300-3-3113-1-1-1000000    orthogonal lifted from C2xA4
ρ1833-1-11-31100-3-311-1-13-1000000    orthogonal lifted from C2xA4
ρ1933-1-1-1-1-130033-1-13-1-1-1000000    orthogonal lifted from A4
ρ2033-1-13-1-1-10033-1-1-13-1-1000000    orthogonal lifted from A4
ρ2133-1-1-1-13-10033-1-1-1-1-13000000    orthogonal lifted from A4
ρ223333111100-3-3-3-3-1-1-1-1000000    orthogonal lifted from C2xA4
ρ236-6-220000006i-6i2i-2i0000000000    complex faithful
ρ246-6-22000000-6i6i-2i2i0000000000    complex faithful

Permutation representations of C4oD4:A4
►On 24 points - transitive group 24T302
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 10 3 12)(2 11 4 9)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)

(1 12)(2 9)(3 10)(4 11)(5 21)(6 22)(7 23)(8 24)(17 19)(18 20)

(1 3)(2 4)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)

(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)
 
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,10,3,12)(2,11,4,9)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,12)(2,9)(3,10)(4,11)(5,21)(6,22)(7,23)(8,24)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9)>;
 
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), (1,10,3,12)(2,11,4,9)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,12)(2,9)(3,10)(4,11)(5,21)(6,22)(7,23)(8,24)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9) );
 
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)], [(1,10,3,12),(2,11,4,9),(5,8,7,6),(13,18,15,20),(14,19,16,17),(21,22,23,24)], [(1,12),(2,9),(3,10),(4,11),(5,21),(6,22),(7,23),(8,24),(17,19),(18,20)], [(1,3),(2,4),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,18,10),(6,19,11),(7,20,12),(8,17,9)]])
 
G:=TransitiveGroup(24,302);
 

Matrix representation of C4oD4:A4 ►in GL5(F13)

50000
05000
001200
000120
000012
,
43000
39000
000112
001012
000012
,
117000
72000
00100
001012
001120
,
10000
01000
001200
001201
001210
,
10000
01000
000121
000120
001120
,
10000
39000
00001
00100
00010

G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,3,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[11,7,0,0,0,7,2,0,0,0,0,0,1,1,1,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,3,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;
 

C4oD4:A4 in GAP, Magma, Sage, TeX 

C_4\circ D_4\rtimes A_4
 
% in TeX
 
G:=Group("C4oD4:A4");
 
// GroupNames label
 
G:=SmallGroup(192,1507);
 
// by ID
 
G=gap.SmallGroup(192,1507);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,262,851,172,1524,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^2=f^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^-1*b*c,c*d=d*c,c*e=e*c,f*c*f^-1=a^-1*b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
 
// generators/relations
 

Export

Character table of C4oD4:A4 in TeX

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