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G = C2×C42⋊C6order 192 = 26·3

Direct product of C2 and C42⋊C6

Aliases: C2×C42⋊C6, C24.4A4, C422(C2×C6), (C2×C42)⋊1C6, C42⋊C33C22, C23.3(C2×A4), C422C22C6, C22.3(C22×A4), (C2×C422C2)⋊C3, (C2×C42⋊C3)⋊1C2, SmallGroup(192,1001)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×C42⋊C6
 Chief series C1 — C22 — C42 — C42⋊C3 — C42⋊C6 — C2×C42⋊C6
 Lower central C42 — C2×C42⋊C6
 Upper central C1 — C2

Generators and relations for C2×C42⋊C6
G = < a,b,c,d | a2=b4=c4=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=c-1, dcd-1=b-1c >

Subgroups: 282 in 66 conjugacy classes, 17 normal (11 characteristic)
C1, C2, C2 [×4], C3, C4 [×4], C22, C22 [×6], C6 [×3], C2×C4 [×8], C23, C23 [×2], C23 [×2], A4, C2×C6, C42, C42, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C24, C2×A4 [×3], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C422C2 [×2], C422C2 [×2], C42⋊C3, C22×A4, C2×C422C2, C2×C42⋊C3, C42⋊C6 [×2], C2×C42⋊C6
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C42⋊C6, C2×C42⋊C6

Character table of C2×C42⋊C6

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F size 1 1 3 3 4 4 16 16 6 6 6 6 12 12 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 trivial ρ2 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 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 linear of order 2 ρ5 1 -1 -1 1 1 -1 ζ32 ζ3 -1 1 1 -1 -1 1 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 linear of order 6 ρ6 1 -1 -1 1 -1 1 ζ32 ζ3 -1 1 1 -1 1 -1 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ7 1 1 1 1 -1 -1 ζ3 ζ32 1 1 1 1 -1 -1 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ8 1 -1 -1 1 -1 1 ζ3 ζ32 -1 1 1 -1 1 -1 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 linear of order 6 ρ9 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ10 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ11 1 -1 -1 1 1 -1 ζ3 ζ32 -1 1 1 -1 -1 1 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 ρ12 1 1 1 1 -1 -1 ζ32 ζ3 1 1 1 1 -1 -1 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 linear of order 6 ρ13 3 3 3 3 3 3 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 -3 -3 3 3 -3 0 0 1 -1 -1 1 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 3 3 -3 -3 0 0 -1 -1 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ16 3 -3 -3 3 -3 3 0 0 1 -1 -1 1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 6 6 -2 -2 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C42⋊C6 ρ18 6 -6 2 -2 0 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 0 0 complex faithful ρ19 6 6 -2 -2 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C42⋊C6 ρ20 6 -6 2 -2 0 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C2×C42⋊C6
On 24 points - transitive group 24T290
Generators in S24
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 17)(2 20 7 23)(3 4 8 12)(5 19)(6 15 11 18)(9 14)(10 22)(13 21 16 24)
(1 22 9 19)(2 11 7 6)(3 16)(4 24)(5 17 10 14)(8 13)(12 21)(15 20 18 23)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,290);

On 24 points - transitive group 24T300
Generators in S24
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 14 7 22)(2 20)(3 6 9 12)(4 17 10 19)(5 15)(8 18)(11 23)(13 24 21 16)
(1 19)(2 11 8 5)(3 24 9 16)(4 14)(6 21 12 13)(7 17)(10 22)(15 20 23 18)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,300);

On 24 points - transitive group 24T306
Generators in S24
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 20 8 18)(2 13 9 21)(4 15 11 23)(5 16 12 24)(6 7)(17 19)
(1 20 8 18)(3 22 10 14)(4 23 11 15)(5 12)(6 17 7 19)(16 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)

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

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

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

G:=TransitiveGroup(24,306);

Matrix representation of C2×C42⋊C6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 5 0 12 8 12 0 0 12
,
 8 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 8 0 0 1 0 4 5 5 0 0 5
,
 7 7 12 0 0 11 5 0 0 11 0 0 0 5 0 0 11 0 11 11 10 0 0 8 12 0 0 8 0 0 3 2 3 6 1 6

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,1,0,12,0,5,0,0,0,8,0,0,1,0,0,12,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[8,0,0,0,0,4,0,12,0,0,8,5,0,0,8,0,0,5,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,5],[7,5,0,11,12,3,7,0,5,11,0,2,12,0,0,10,0,3,0,11,0,0,8,6,0,0,11,0,0,1,11,0,0,8,0,6] >;

C2×C42⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes C_6
% in TeX

G:=Group("C2xC4^2:C6");
// GroupNames label

G:=SmallGroup(192,1001);
// by ID

G=gap.SmallGroup(192,1001);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,1683,185,360,4204,1173,102,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=c^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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