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## G = C2×C42⋊C3order 96 = 25·3

### Direct product of C2 and C42⋊C3

Aliases: C2×C42⋊C3, C423C6, C23.4A4, (C2×C42)⋊C3, C22.1(C2×A4), SmallGroup(96,68)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C42⋊C3
G = < a,b,c,d | a2=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

Character table of C2×C42⋊C3

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B size 1 1 3 3 16 16 3 3 3 3 3 3 3 3 16 16 ρ1 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 linear of order 2 ρ3 1 -1 -1 1 ζ3 ζ32 1 1 1 1 -1 -1 -1 -1 ζ6 ζ65 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 linear of order 3 ρ6 1 -1 -1 1 ζ32 ζ3 1 1 1 1 -1 -1 -1 -1 ζ65 ζ6 linear of order 6 ρ7 3 -3 -3 3 0 0 -1 -1 -1 -1 1 1 1 1 0 0 orthogonal lifted from C2×A4 ρ8 3 3 3 3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ9 3 3 -1 -1 0 0 -1-2i 1 -1+2i 1 -1-2i 1 1 -1+2i 0 0 complex lifted from C42⋊C3 ρ10 3 3 -1 -1 0 0 1 -1-2i 1 -1+2i 1 -1+2i -1-2i 1 0 0 complex lifted from C42⋊C3 ρ11 3 3 -1 -1 0 0 1 -1+2i 1 -1-2i 1 -1-2i -1+2i 1 0 0 complex lifted from C42⋊C3 ρ12 3 -3 1 -1 0 0 -1+2i 1 -1-2i 1 1-2i -1 -1 1+2i 0 0 complex faithful ρ13 3 -3 1 -1 0 0 1 -1+2i 1 -1-2i -1 1+2i 1-2i -1 0 0 complex faithful ρ14 3 -3 1 -1 0 0 -1-2i 1 -1+2i 1 1+2i -1 -1 1-2i 0 0 complex faithful ρ15 3 -3 1 -1 0 0 1 -1-2i 1 -1+2i -1 1-2i 1+2i -1 0 0 complex faithful ρ16 3 3 -1 -1 0 0 -1+2i 1 -1-2i 1 -1+2i 1 1 -1-2i 0 0 complex lifted from C42⋊C3

Permutation representations of C2×C42⋊C3
On 12 points - transitive group 12T55
Generators in S12
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 2 3 4)(5 7)(6 8)(9 10 11 12)
(1 7 12)(2 6 9)(3 5 10)(4 8 11)

G:=sub<Sym(12)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,3,4)(5,7)(6,8)(9,10,11,12), (1,7,12)(2,6,9)(3,5,10)(4,8,11)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,3,4)(5,7)(6,8)(9,10,11,12), (1,7,12)(2,6,9)(3,5,10)(4,8,11) );

G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(5,6,7,8),(9,10,11,12)], [(1,2,3,4),(5,7),(6,8),(9,10,11,12)], [(1,7,12),(2,6,9),(3,5,10),(4,8,11)]])

G:=TransitiveGroup(12,55);

On 24 points - transitive group 24T173
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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 6 7 3)(2 5 8 4)(9 14 11 16)(10 15 12 13)(17 21)(18 22)(19 23)(20 24)
(1 21 10)(2 19 14)(3 20 9)(4 22 13)(5 24 15)(6 18 11)(7 23 12)(8 17 16)

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

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

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

G:=TransitiveGroup(24,173);

On 24 points - transitive group 24T174
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 14)(10 15)(11 16)(12 13)(17 24)(18 21)(19 22)(20 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)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 16 15 14)
(1 19 15)(2 17 13)(3 18 14)(4 20 16)(5 22 10)(6 24 12)(7 21 9)(8 23 11)

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

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

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

G:=TransitiveGroup(24,174);

On 24 points - transitive group 24T175
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(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 8 3 6)(2 7 4 5)(9 21 11 23)(10 22 12 24)(13 19)(14 20)(15 17)(16 18)
(1 17 11)(2 13 22)(3 19 9)(4 15 24)(5 18 21)(6 14 10)(7 20 23)(8 16 12)

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

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

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

G:=TransitiveGroup(24,175);

C2×C42⋊C3 is a maximal subgroup of   C23.9S4  C42⋊C12  C422C12
C2×C42⋊C3 is a maximal quotient of   C424C4⋊C3

Polynomial with Galois group C2×C42⋊C3 over ℚ
actionf(x)Disc(f)
12T55x12-38x10+538x8-3458x6+9659x4-8788x2+169224·312·712·1314·2234

Matrix representation of C2×C42⋊C3 in GL3(𝔽5) generated by

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

C2×C42⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes C_3
% in TeX

G:=Group("C2xC4^2:C3");
// GroupNames label

G:=SmallGroup(96,68);
// by ID

G=gap.SmallGroup(96,68);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,116,230,801,69,730,1307]);
// Polycyclic

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

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