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G = C42⋊C3order 48 = 24·3

The semidirect product of C42 and C3 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C42⋊C3, C22.A4, SmallGroup(48,3)

Series: Derived Chief Lower central Upper central

C1C42 — C42⋊C3
C1C22C42 — C42⋊C3
C42 — C42⋊C3
C1

Generators and relations for C42⋊C3
 G = < a,b,c | a4=b4=c3=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >

3C2
16C3
3C4
3C4
3C2×C4
4A4

Character table of C42⋊C3

 class 123A3B4A4B4C4D
 size 1316163333
ρ111111111    trivial
ρ211ζ32ζ31111    linear of order 3
ρ311ζ3ζ321111    linear of order 3
ρ43300-1-1-1-1    orthogonal lifted from A4
ρ53-1001-1+2i-1-2i1    complex faithful
ρ63-100-1-2i11-1+2i    complex faithful
ρ73-1001-1-2i-1+2i1    complex faithful
ρ83-100-1+2i11-1-2i    complex faithful

Permutation representations of C42⋊C3
On 12 points - transitive group 12T31
Generators in S12
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 8 7 6)
(1 11 5)(2 9 7)(3 10 8)(4 12 6)

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

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

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

G:=TransitiveGroup(12,31);

On 16 points - transitive group 16T63
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)

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

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

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

G:=TransitiveGroup(16,63);

On 24 points - transitive group 24T58
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 7 6 4)(2 8 5 3)(9 19)(10 20)(11 17)(12 18)(13 22 15 24)(14 23 16 21)
(1 12 24)(2 20 16)(3 9 15)(4 17 23)(5 18 14)(6 10 22)(7 19 21)(8 11 13)

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

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

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

G:=TransitiveGroup(24,58);

Polynomial with Galois group C42⋊C3 over ℚ
actionf(x)Disc(f)
12T31x12-38x10+450x8-2106x6+3679x4-2028x2+169224·720·1314·7974

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

200
020
004
,
100
030
002
,
001
100
010
G:=sub<GL(3,GF(5))| [2,0,0,0,2,0,0,0,4],[1,0,0,0,3,0,0,0,2],[0,1,0,0,0,1,1,0,0] >;

C42⋊C3 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_3
% in TeX

G:=Group("C4^2:C3");
// GroupNames label

G:=SmallGroup(48,3);
// by ID

G=gap.SmallGroup(48,3);
# by ID

G:=PCGroup([5,-3,-2,2,-2,2,61,126,497,42,483,904]);
// Polycyclic

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

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