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G = C4.Dic3order 48 = 24·3

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

metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C6 — C4.Dic3
C1C3C6C12C3⋊C8 — C4.Dic3
C3C6 — C4.Dic3
C1C4C2×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 >

2C2
2C6
3C8
3C8
3M4(2)

Character table of C4.Dic3

 class 12A2B34A4B4C6A6B6C8A8B8C8D12A12B12C12D
 size 112211222266662222
ρ1111111111111111111    trivial
ρ211-1111-1-1-111-1-111-1-11    linear of order 2
ρ31111111111-1-1-1-11111    linear of order 2
ρ411-1111-1-1-11-111-11-1-11    linear of order 2
ρ51111-1-1-1111-ii-ii-1-1-1-1    linear of order 4
ρ611-11-1-11-1-11ii-i-i-111-1    linear of order 4
ρ711-11-1-11-1-11-i-iii-111-1    linear of order 4
ρ81111-1-1-1111i-ii-i-1-1-1-1    linear of order 4
ρ9222-1222-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ1022-2-122-211-10000-111-1    orthogonal lifted from D6
ρ11222-1-2-2-2-1-1-100001111    symplectic lifted from Dic3, Schur index 2
ρ1222-2-1-2-2211-100001-1-11    symplectic lifted from Dic3, Schur index 2
ρ132-202-2i2i000-200002i00-2i    complex lifted from M4(2)
ρ142-2022i-2i000-20000-2i002i    complex lifted from M4(2)
ρ152-20-1-2i2i0--3-310000ζ2-33ζ2    complex faithful
ρ162-20-12i-2i0-3--310000ζ2-33ζ2    complex faithful
ρ172-20-12i-2i0--3-310000ζ23-3ζ2    complex faithful
ρ182-20-1-2i2i0-3--310000ζ23-3ζ2    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 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)

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

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

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

G:=TransitiveGroup(24,20);

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

80
05
,
20
06
,
05
10
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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