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## G = C42⋊4C4⋊C3order 192 = 26·3

### The semidirect product of C42⋊4C4 and C3 acting faithfully

Aliases: C424C4⋊C3, C4.1(C42⋊C3), (C22×C4).8A4, C23.10(C2×A4), C22.1(C4.A4), C2.C42.3C6, C23.3A4.3C2, C2.3(C2×C42⋊C3), SmallGroup(192,190)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C42⋊4C4⋊C3
 Chief series C1 — C2 — C23 — C2.C42 — C23.3A4 — C42⋊4C4⋊C3
 Lower central C2.C42 — C42⋊4C4⋊C3
 Upper central C1 — C4

Generators and relations for C424C4⋊C3
G = < a,b,c,d | a4=b4=c4=d3=1, ab=ba, cac-1=ab2, dad-1=a2c-1, bc=cb, dbd-1=a2b-1, dcd-1=a-1b2c >

Character table of C424C4⋊C3

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 12A 12B 12C 12D size 1 1 3 3 16 16 1 1 3 3 6 6 6 6 6 6 6 6 16 16 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ5 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ7 2 -2 2 -2 -1 -1 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 1 1 -i -i i i complex lifted from C4.A4 ρ8 2 -2 2 -2 -1 -1 2i -2i 2i -2i 0 0 0 0 0 0 0 0 1 1 i i -i -i complex lifted from C4.A4 ρ9 2 -2 2 -2 ζ65 ζ6 2i -2i 2i -2i 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 complex lifted from C4.A4 ρ10 2 -2 2 -2 ζ6 ζ65 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 complex lifted from C4.A4 ρ11 2 -2 2 -2 ζ6 ζ65 2i -2i 2i -2i 0 0 0 0 0 0 0 0 ζ32 ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 complex lifted from C4.A4 ρ12 2 -2 2 -2 ζ65 ζ6 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 ζ3 ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 complex lifted from C4.A4 ρ13 3 3 3 3 0 0 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 3 3 3 0 0 -3 -3 -3 -3 -1 -1 -1 -1 1 1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 -1 -1 0 0 3 3 -1 -1 -1-2i -1+2i 1 1 -1-2i 1 1 -1+2i 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ16 3 3 -1 -1 0 0 -3 -3 1 1 1 1 -1+2i -1-2i -1 1-2i 1+2i -1 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ17 3 3 -1 -1 0 0 3 3 -1 -1 1 1 -1+2i -1-2i 1 -1+2i -1-2i 1 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ18 3 3 -1 -1 0 0 -3 -3 1 1 1 1 -1-2i -1+2i -1 1+2i 1-2i -1 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ19 3 3 -1 -1 0 0 3 3 -1 -1 -1+2i -1-2i 1 1 -1+2i 1 1 -1-2i 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ20 3 3 -1 -1 0 0 3 3 -1 -1 1 1 -1-2i -1+2i 1 -1-2i -1+2i 1 0 0 0 0 0 0 complex lifted from C42⋊C3 ρ21 3 3 -1 -1 0 0 -3 -3 1 1 -1+2i -1-2i 1 1 1-2i -1 -1 1+2i 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ22 3 3 -1 -1 0 0 -3 -3 1 1 -1-2i -1+2i 1 1 1+2i -1 -1 1-2i 0 0 0 0 0 0 complex lifted from C2×C42⋊C3 ρ23 6 -6 -2 2 0 0 6i -6i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 6 -6 -2 2 0 0 -6i 6i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C424C4⋊C3
On 24 points - transitive group 24T301
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 16 13 12)(2 15 14 11)(3 9 5 8)(4 10 6 7)(17 18 19 20)(21 24 23 22)
(1 12 13 16)(2 15 14 11)(3 7)(4 9)(5 10)(6 8)(17 21)(18 24)(19 23)(20 22)
(1 24 6)(2 17 8)(3 11 18)(4 13 22)(5 15 20)(7 12 23)(9 14 19)(10 16 21)

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

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

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

G:=TransitiveGroup(24,301);

On 24 points - transitive group 24T309
Generators in S24
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 2 4 3)(5 12 6 11)(7 10 13 15)(8 9 14 16)(17 22 19 24)(18 23 20 21)
(1 11)(2 5)(3 6)(4 12)(7 8 13 14)(9 15 16 10)(17 19)(22 24)
(1 8 17)(2 16 22)(3 9 24)(4 14 19)(5 15 23)(6 10 21)(7 18 11)(12 13 20)

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

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

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

G:=TransitiveGroup(24,309);

Matrix representation of C424C4⋊C3 in GL5(𝔽13)

 9 3 0 0 0 3 4 0 0 0 0 0 5 0 0 0 0 0 8 0 0 0 0 0 1
,
 5 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1
,
 10 9 0 0 0 9 3 0 0 0 0 0 8 0 0 0 0 0 12 0 0 0 0 0 8
,
 1 0 0 0 0 9 3 0 0 0 0 0 0 0 3 0 0 3 0 0 0 0 0 3 0

G:=sub<GL(5,GF(13))| [9,3,0,0,0,3,4,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,1],[5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1],[10,9,0,0,0,9,3,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,8],[1,9,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0] >;

C424C4⋊C3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4C_4\rtimes C_3
% in TeX

G:=Group("C4^2:4C4:C3");
// GroupNames label

G:=SmallGroup(192,190);
// by ID

G=gap.SmallGroup(192,190);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,268,934,521,80,2531,3540]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=d^3=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^2*c^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*b^2*c>;
// generators/relations

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