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G = C4○D4⋊A4  order 192 = 26·3

1st semidirect product of C4○D4 and A4 acting via A4/C22=C3

non-abelian, soluble

Aliases: C4○D4⋊1A4, C22⋊(C4.A4), Q8.1(C2×A4), Q8⋊A4⋊3C2, (C22×C4).9A4, C4.1(C22⋊A4), C23.23(C2×A4), (C22×Q8).7C6, (C22×C4○D4)⋊3C3, C2.3(C2×C22⋊A4), SmallGroup(192,1507)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C22×Q8 — C4○D4⋊A4
Chief series
C1 — C2 — Q8 — C22×Q8 — Q8⋊A4 — C4○D4⋊A4
Lower central
C22×Q8 — C4○D4⋊A4

Subgroups: 599 in 175 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×5], C22, C22 [×18], C6, C2×C4 [×24], D4 [×16], Q8 [×4], Q8 [×4], C23, C23 [×10], C12, A4, C22×C4, C22×C4 [×13], C2×D4 [×12], C2×Q8 [×4], C4○D4 [×4], C4○D4 [×20], C24, SL2(𝔽3) [×4], C2×A4, C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×8], C4×A4, C4.A4 [×4], C22×C4○D4, Q8⋊A4, C4○D4⋊A4

Quotients:
C1, C2, C3, C6, A4 [×5], C2×A4 [×5], C4.A4, C22⋊A4, C2×C22⋊A4, C4○D4⋊A4

Generators and relations
 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 >

Permutation representations
►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 20 15 18)(14 17 16 19)(21 22 23 24)

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

Matrix representation ►G ⊆ 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] >;
 

Character table of C4○D4⋊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
ρ72-22-20000-1-12i2i2i2i000011-i-iii    complex lifted from C4.A4
ρ82-22-20000-1-12i2i2i2i000011ii-i-i    complex lifted from C4.A4
ρ92-22-20000ζ6ζ652i2i2i2i0000ζ32ζ62ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    complex lifted from C4.A4
ρ102-22-20000ζ65ζ62i2i2i2i0000ζ62ζ32ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    complex lifted from C4.A4
ρ112-22-20000ζ65ζ62i2i2i2i0000ζ62ζ32ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    complex lifted from C4.A4
ρ122-22-20000ζ6ζ652i2i2i2i0000ζ32ζ62ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    complex lifted from C4.A4
ρ1333-1-111-3100-3-311-1-1-13000000    orthogonal lifted from C2×A4
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 C2×A4
ρ1733-1-1111-300-3-3113-1-1-1000000    orthogonal lifted from C2×A4
ρ1833-1-11-31100-3-311-1-13-1000000    orthogonal lifted from C2×A4
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 C2×A4
ρ236-6-220000006i6i2i2i0000000000    complex faithful
ρ246-6-220000006i6i2i2i0000000000    complex faithful

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
 

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