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G = C4.A4order 48 = 24·3

The central extension by C4 of A4

non-abelian, soluble

Aliases: C4.A4, Q8.C6, C4SL2(𝔽3), SL2(𝔽3)⋊2C2, C4○D4⋊C3, C2.3(C2×A4), SmallGroup(48,33)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C4.A4
C1C2Q8SL2(𝔽3) — C4.A4
Q8 — C4.A4
C1C4

Generators and relations for C4.A4
 G = < a,b,c,d | a4=d3=1, b2=c2=a2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=a2bc, dcd-1=b >

6C2
4C3
3C22
3C4
4C6
3D4
3C2×C4
4C12

Character table of C4.A4

 class 12A2B3A3B4A4B4C6A6B12A12B12C12D
 size 11644116444444
ρ111111111111111    trivial
ρ211-111-1-1111-1-1-1-1    linear of order 2
ρ3111ζ32ζ3111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ411-1ζ32ζ3-1-11ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ5111ζ3ζ32111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ611-1ζ3ζ32-1-11ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ72-20-1-12i-2i011ii-i-i    complex faithful
ρ82-20-1-1-2i2i011-i-iii    complex faithful
ρ92-20ζ65ζ6-2i2i0ζ32ζ62ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    complex faithful
ρ102-20ζ65ζ62i-2i0ζ32ζ62ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    complex faithful
ρ112-20ζ6ζ652i-2i0ζ62ζ32ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    complex faithful
ρ122-20ζ6ζ65-2i2i0ζ62ζ32ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    complex faithful
ρ1333100-3-3-1000000    orthogonal lifted from C2×A4
ρ1433-10033-1000000    orthogonal lifted from A4

Permutation representations of C4.A4
On 16 points - transitive group 16T60
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 3 8)(2 7 4 5)(9 13 11 15)(10 14 12 16)
(1 15 3 13)(2 16 4 14)(5 10 7 12)(6 11 8 9)
(5 14 12)(6 15 9)(7 16 10)(8 13 11)

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

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

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

G:=TransitiveGroup(16,60);

On 24 points - transitive group 24T21
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 4 3 2)(5 23 7 21)(6 24 8 22)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 12 3 10)(2 9 4 11)(5 6 7 8)(13 18 15 20)(14 19 16 17)(21 24 23 22)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)

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

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

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

G:=TransitiveGroup(24,21);

Matrix representation of C4.A4 in GL2(𝔽5) generated by

20
02
,
33
02
,
20
13
,
04
14
G:=sub<GL(2,GF(5))| [2,0,0,2],[3,0,3,2],[2,1,0,3],[0,1,4,4] >;

C4.A4 in GAP, Magma, Sage, TeX

C_4.A_4
% in TeX

G:=Group("C4.A4");
// GroupNames label

G:=SmallGroup(48,33);
// by ID

G=gap.SmallGroup(48,33);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-2,120,97,72,188,133,58]);
// Polycyclic

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

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