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## G = C2×C3.A4order 72 = 23·32

### Direct product of C2 and C3.A4

Aliases: C2×C3.A4, C23⋊C9, C22⋊C18, C6.2A4, C3.(C2×A4), (C2×C6).C6, (C22×C6).C3, SmallGroup(72,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C3.A4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4
 Lower central C22 — C2×C3.A4
 Upper central C1 — C6

Generators and relations for C2×C3.A4
G = < a,b,c,d,e | a2=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Character table of C2×C3.A4

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 9A 9B 9C 9D 9E 9F 18A 18B 18C 18D 18E 18F size 1 1 3 3 1 1 1 1 3 3 3 3 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 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 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ5 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ6 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ7 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ9 ζ97 ζ92 ζ95 ζ98 ζ94 ζ9 ζ94 ζ97 ζ92 ζ95 ζ98 linear of order 9 ρ8 1 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 ζ95 ζ98 ζ9 ζ97 ζ94 ζ92 -ζ95 -ζ92 -ζ98 -ζ9 -ζ97 -ζ94 linear of order 18 ρ9 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ94 ζ9 ζ98 ζ92 ζ95 ζ97 ζ94 ζ97 ζ9 ζ98 ζ92 ζ95 linear of order 9 ρ10 1 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 ζ9 ζ97 ζ92 ζ95 ζ98 ζ94 -ζ9 -ζ94 -ζ97 -ζ92 -ζ95 -ζ98 linear of order 18 ρ11 1 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 ζ97 ζ94 ζ95 ζ98 ζ92 ζ9 -ζ97 -ζ9 -ζ94 -ζ95 -ζ98 -ζ92 linear of order 18 ρ12 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ95 ζ98 ζ9 ζ97 ζ94 ζ92 ζ95 ζ92 ζ98 ζ9 ζ97 ζ94 linear of order 9 ρ13 1 -1 -1 1 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 ζ94 ζ9 ζ98 ζ92 ζ95 ζ97 -ζ94 -ζ97 -ζ9 -ζ98 -ζ92 -ζ95 linear of order 18 ρ14 1 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 ζ92 ζ95 ζ94 ζ9 ζ97 ζ98 -ζ92 -ζ98 -ζ95 -ζ94 -ζ9 -ζ97 linear of order 18 ρ15 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ92 ζ95 ζ94 ζ9 ζ97 ζ98 ζ92 ζ98 ζ95 ζ94 ζ9 ζ97 linear of order 9 ρ16 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ98 ζ92 ζ97 ζ94 ζ9 ζ95 ζ98 ζ95 ζ92 ζ97 ζ94 ζ9 linear of order 9 ρ17 1 -1 -1 1 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 ζ98 ζ92 ζ97 ζ94 ζ9 ζ95 -ζ98 -ζ95 -ζ92 -ζ97 -ζ94 -ζ9 linear of order 18 ρ18 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ97 ζ94 ζ95 ζ98 ζ92 ζ9 ζ97 ζ9 ζ94 ζ95 ζ98 ζ92 linear of order 9 ρ19 3 3 -1 -1 3 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ20 3 -3 1 -1 3 3 -3 -3 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ21 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.A4 ρ22 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 3+3√-3/2 3-3√-3/2 ζ6 ζ65 ζ3 ζ32 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ23 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 3-3√-3/2 3+3√-3/2 ζ65 ζ6 ζ32 ζ3 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3.A4

Permutation representations of C2×C3.A4
On 18 points - transitive group 18T26
Generators in S18
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(2 17)(3 18)(5 11)(6 12)(8 14)(9 15)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,26);

C2×C3.A4 is a maximal subgroup of
C6.S4  A4×C18  C2.(C42⋊C9)  C24⋊C18  C42⋊C18  C422C18  C22⋊(Q8⋊C9)  2+ 1+42C9
C2×C3.A4 is a maximal quotient of
Q8.C18  C24⋊C18  C42⋊C18  C422C18

Matrix representation of C2×C3.A4 in GL3(𝔽7) generated by

 6 0 0 0 6 0 0 0 6
,
 4 0 0 0 4 0 0 0 4
,
 6 0 0 0 1 0 0 0 6
,
 6 0 0 0 6 0 0 0 1
,
 0 0 4 6 0 0 0 6 0
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,0,0,0,4,0,0,0,4],[6,0,0,0,1,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,6,0,0,0,6,4,0,0] >;

C2×C3.A4 in GAP, Magma, Sage, TeX

C_2\times C_3.A_4
% in TeX

G:=Group("C2xC3.A4");
// GroupNames label

G:=SmallGroup(72,16);
// by ID

G=gap.SmallGroup(72,16);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,2,36,368,684]);
// Polycyclic

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

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