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## G = C2×C4.A4order 96 = 25·3

### Direct product of C2 and C4.A4

Aliases: C2×C4.A4, SL2(𝔽3)⋊3C22, C4(C4.A4), C4○D42C6, C4.6(C2×A4), (C2×C4).2A4, Q8.1(C2×C6), (C2×Q8).2C6, C22.9(C2×A4), C2.5(C22×A4), (C2×C4)SL2(𝔽3), C4(C2×SL2(𝔽3)), (C2×SL2(𝔽3))⋊4C2, (C2×C4○D4)⋊C3, (C2×C4)(C2×SL2(𝔽3)), SmallGroup(96,200)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×C4.A4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×C4.A4
 Lower central Q8 — C2×C4.A4
 Upper central C1 — C2×C4

Generators and relations for C2×C4.A4
G = < a,b,c,d,e | a2=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Character table of C2×C4.A4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 6 6 4 4 1 1 1 1 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 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 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 -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 -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 -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 ζ32 ζ32 ζ32 ζ3 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ7 1 -1 1 -1 1 -1 ζ3 ζ32 1 -1 1 -1 1 -1 ζ32 ζ65 ζ65 ζ3 ζ6 ζ6 ζ65 ζ65 ζ6 ζ6 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ8 1 -1 1 -1 -1 1 ζ3 ζ32 -1 1 -1 1 1 -1 ζ32 ζ65 ζ65 ζ3 ζ6 ζ6 ζ3 ζ3 ζ32 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ9 1 -1 1 -1 -1 1 ζ32 ζ3 -1 1 -1 1 1 -1 ζ3 ζ6 ζ6 ζ32 ζ65 ζ65 ζ32 ζ32 ζ3 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ10 1 -1 1 -1 1 -1 ζ32 ζ3 1 -1 1 -1 1 -1 ζ3 ζ6 ζ6 ζ32 ζ65 ζ65 ζ6 ζ6 ζ65 ζ65 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ11 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ12 1 1 1 1 -1 -1 ζ3 ζ32 -1 -1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ13 2 -2 -2 2 0 0 -1 -1 2i -2i -2i 2i 0 0 1 1 -1 1 1 -1 -i i -i i i -i i -i complex lifted from C4.A4 ρ14 2 2 -2 -2 0 0 -1 -1 -2i -2i 2i 2i 0 0 1 -1 1 1 -1 1 -i i -i i -i i -i i complex lifted from C4.A4 ρ15 2 -2 -2 2 0 0 -1 -1 -2i 2i 2i -2i 0 0 1 1 -1 1 1 -1 i -i i -i -i i -i i complex lifted from C4.A4 ρ16 2 2 -2 -2 0 0 -1 -1 2i 2i -2i -2i 0 0 1 -1 1 1 -1 1 i -i i -i i -i i -i complex lifted from C4.A4 ρ17 2 -2 -2 2 0 0 ζ6 ζ65 2i -2i -2i 2i 0 0 ζ3 ζ32 ζ6 ζ32 ζ3 ζ65 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex lifted from C4.A4 ρ18 2 2 -2 -2 0 0 ζ65 ζ6 -2i -2i 2i 2i 0 0 ζ32 ζ65 ζ3 ζ3 ζ6 ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex lifted from C4.A4 ρ19 2 -2 -2 2 0 0 ζ65 ζ6 2i -2i -2i 2i 0 0 ζ32 ζ3 ζ65 ζ3 ζ32 ζ6 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex lifted from C4.A4 ρ20 2 -2 -2 2 0 0 ζ6 ζ65 -2i 2i 2i -2i 0 0 ζ3 ζ32 ζ6 ζ32 ζ3 ζ65 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex lifted from C4.A4 ρ21 2 2 -2 -2 0 0 ζ6 ζ65 2i 2i -2i -2i 0 0 ζ3 ζ6 ζ32 ζ32 ζ65 ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex lifted from C4.A4 ρ22 2 -2 -2 2 0 0 ζ65 ζ6 -2i 2i 2i -2i 0 0 ζ32 ζ3 ζ65 ζ3 ζ32 ζ6 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex lifted from C4.A4 ρ23 2 2 -2 -2 0 0 ζ65 ζ6 2i 2i -2i -2i 0 0 ζ32 ζ65 ζ3 ζ3 ζ6 ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex lifted from C4.A4 ρ24 2 2 -2 -2 0 0 ζ6 ζ65 -2i -2i 2i 2i 0 0 ζ3 ζ6 ζ32 ζ32 ζ65 ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex lifted from C4.A4 ρ25 3 -3 3 -3 -1 1 0 0 3 -3 3 -3 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ26 3 3 3 3 -1 -1 0 0 3 3 3 3 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ27 3 -3 3 -3 1 -1 0 0 -3 3 -3 3 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ28 3 3 3 3 1 1 0 0 -3 -3 -3 -3 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4

Smallest permutation representation of C2×C4.A4
On 32 points
Generators in S32
(1 13)(2 14)(3 15)(4 16)(5 26)(6 27)(7 28)(8 25)(9 17)(10 18)(11 19)(12 20)(21 32)(22 29)(23 30)(24 31)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 27 3 25)(2 28 4 26)(5 14 7 16)(6 15 8 13)(9 31 11 29)(10 32 12 30)(17 24 19 22)(18 21 20 23)
(1 29 3 31)(2 30 4 32)(5 18 7 20)(6 19 8 17)(9 27 11 25)(10 28 12 26)(13 22 15 24)(14 23 16 21)
(5 21 20)(6 22 17)(7 23 18)(8 24 19)(9 27 29)(10 28 30)(11 25 31)(12 26 32)

G:=sub<Sym(32)| (1,13)(2,14)(3,15)(4,16)(5,26)(6,27)(7,28)(8,25)(9,17)(10,18)(11,19)(12,20)(21,32)(22,29)(23,30)(24,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,27,3,25)(2,28,4,26)(5,14,7,16)(6,15,8,13)(9,31,11,29)(10,32,12,30)(17,24,19,22)(18,21,20,23), (1,29,3,31)(2,30,4,32)(5,18,7,20)(6,19,8,17)(9,27,11,25)(10,28,12,26)(13,22,15,24)(14,23,16,21), (5,21,20)(6,22,17)(7,23,18)(8,24,19)(9,27,29)(10,28,30)(11,25,31)(12,26,32)>;

G:=Group( (1,13)(2,14)(3,15)(4,16)(5,26)(6,27)(7,28)(8,25)(9,17)(10,18)(11,19)(12,20)(21,32)(22,29)(23,30)(24,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,27,3,25)(2,28,4,26)(5,14,7,16)(6,15,8,13)(9,31,11,29)(10,32,12,30)(17,24,19,22)(18,21,20,23), (1,29,3,31)(2,30,4,32)(5,18,7,20)(6,19,8,17)(9,27,11,25)(10,28,12,26)(13,22,15,24)(14,23,16,21), (5,21,20)(6,22,17)(7,23,18)(8,24,19)(9,27,29)(10,28,30)(11,25,31)(12,26,32) );

G=PermutationGroup([(1,13),(2,14),(3,15),(4,16),(5,26),(6,27),(7,28),(8,25),(9,17),(10,18),(11,19),(12,20),(21,32),(22,29),(23,30),(24,31)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,27,3,25),(2,28,4,26),(5,14,7,16),(6,15,8,13),(9,31,11,29),(10,32,12,30),(17,24,19,22),(18,21,20,23)], [(1,29,3,31),(2,30,4,32),(5,18,7,20),(6,19,8,17),(9,27,11,25),(10,28,12,26),(13,22,15,24),(14,23,16,21)], [(5,21,20),(6,22,17),(7,23,18),(8,24,19),(9,27,29),(10,28,30),(11,25,31),(12,26,32)])

C2×C4.A4 is a maximal subgroup of
U2(𝔽3)⋊C2  C4.A4⋊C4  SL2(𝔽3).D4  (C2×C4).S4  SL2(𝔽3)⋊D4  C4○D4⋊C12  SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  M4(2).A4  GL2(𝔽3)⋊C22  2- 1+43C6
C2×C4.A4 is a maximal quotient of
C2×C4×SL2(𝔽3)  SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  SL2(𝔽3)⋊3Q8

Matrix representation of C2×C4.A4 in GL3(𝔽13) generated by

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

C2×C4.A4 in GAP, Magma, Sage, TeX

C_2\times C_4.A_4
% in TeX

G:=Group("C2xC4.A4");
// GroupNames label

G:=SmallGroup(96,200);
// by ID

G=gap.SmallGroup(96,200);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,295,159,117,286,202,88]);
// Polycyclic

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

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