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G = C4.Dic3  order 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, C3⋊2M4(2), C22.Dic3, C12.15C22, C3⋊C8⋊5C2, (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 >

C1
C2
2C2
C3
C4
C22
C4
C6
2C6
C2×C4
3C8
3C8
C2×C6
C12
C12
3M4(2)
C3⋊C8
C3⋊C8
C2×C12
C4.Dic3

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√-310000-i-√3√3i    complex faithful
ρ162-20-12i-2i0√-3-√-310000i-√3√3-i    complex faithful
ρ172-20-12i-2i0-√-3√-310000i√3-√3-i    complex faithful
ρ182-20-1-2i2i0√-3-√-310000-i√3-√3i    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);
 

C4.Dic3 is a maximal subgroup of
 C42⋊4S3  C24.C4  C12.53D4  C12.46D4  C12.47D4  C12.D4  C12.10D4  Q8⋊3Dic3  C8○D12  S3×M4(2)  D12⋊6C22  Q8.11D6  D4.Dic3  D4⋊D6  Q8.14D6  C4.Dic9  D6.Dic3  C12.58D6  A4⋊M4(2)  U2(𝔽3)⋊C2  C20.32D6  C60.7C4  C12.F5  C15⋊8M4(2)  C28.32D6  C84.C4  C33⋊4M4(2)  C33⋊12M4(2)
C4.Dic3 is a maximal quotient of
 C42.S3  C12⋊C8  C12.55D4  C4.Dic9  D6.Dic3  C12.58D6  A4⋊M4(2)  C20.32D6  C60.7C4  C12.F5  C15⋊8M4(2)  C28.32D6  C84.C4  C33⋊4M4(2)  C33⋊12M4(2)

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
 

Export

Subgroup lattice of C4.Dic3 in TeX
Character table of C4.Dic3 in TeX

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