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G = C4.Dic3order 48 = 24·3

The non-split extension by C4 of Dic3 acting via Dic3/C6=C2

Aliases: C4.Dic3, C12.1C4, C4.15D6, C32M4(2), C22.Dic3, C12.15C22, C3⋊C85C2, (C2×C6).3C4, (C2×C4).2S3, C6.6(C2×C4), (C2×C12).4C2, C2.3(C2×Dic3), SmallGroup(48,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C4.Dic3
 Lower central C3 — C6 — C4.Dic3
 Upper central C1 — C4 — C2×C4

Generators and relations for C4.Dic3
G = < a,b,c | a4=1, b6=a2, c2=a2b3, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C4.Dic3

 class 1 2A 2B 3 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 2 2 1 1 2 2 2 2 6 6 6 6 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 1 1 1 -i i -i i -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 -1 -1 1 i i -i -i -1 1 1 -1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 -1 -1 1 -i -i i i -1 1 1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 1 1 i -i i -i -1 -1 -1 -1 linear of order 4 ρ9 2 2 2 -1 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -1 2 2 -2 1 1 -1 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 -1 -2 -2 -2 -1 -1 -1 0 0 0 0 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 2 -2 -1 -2 -2 2 1 1 -1 0 0 0 0 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ13 2 -2 0 2 -2i 2i 0 0 0 -2 0 0 0 0 2i 0 0 -2i complex lifted from M4(2) ρ14 2 -2 0 2 2i -2i 0 0 0 -2 0 0 0 0 -2i 0 0 2i complex lifted from M4(2) ρ15 2 -2 0 -1 -2i 2i 0 -√-3 √-3 1 0 0 0 0 -i -√3 √3 i complex faithful ρ16 2 -2 0 -1 2i -2i 0 √-3 -√-3 1 0 0 0 0 i -√3 √3 -i complex faithful ρ17 2 -2 0 -1 2i -2i 0 -√-3 √-3 1 0 0 0 0 i √3 -√3 -i complex faithful ρ18 2 -2 0 -1 -2i 2i 0 √-3 -√-3 1 0 0 0 0 -i √3 -√3 i complex faithful

Permutation representations of C4.Dic3
On 24 points - transitive group 24T20
Generators in S24
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 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 16 10 13 7 22 4 19)(2 21 11 18 8 15 5 24)(3 14 12 23 9 20 6 17)

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

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

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

G:=TransitiveGroup(24,20);

Matrix representation of C4.Dic3 in GL2(𝔽13) generated by

 8 0 0 5
,
 2 0 0 6
,
 0 5 1 0
G:=sub<GL(2,GF(13))| [8,0,0,5],[2,0,0,6],[0,1,5,0] >;

C4.Dic3 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_3
% in TeX

G:=Group("C4.Dic3");
// GroupNames label

G:=SmallGroup(48,10);
// by ID

G=gap.SmallGroup(48,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,20,101,42,804]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^6=a^2,c^2=a^2*b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

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