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G = Dic12  order 48 = 24·3

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic12, C8.S3, C3⋊1Q16, C6.3D4, C24.1C2, C4.10D6, C2.5D12, Dic6.1C2, C12.10C22, SmallGroup(48,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — Dic12
Chief series
C1 — C3 — C6 — C12 — Dic6 — Dic12
Lower central
C3 — C6 — C12 — Dic12
Upper central
C1 — C2 — C4 — C8

Generators and relations for Dic12
 G = < a,b | a24=1, b2=a12, bab-1=a-1 >

C1
C2
C3
C4
6C4
6C4
C6
C8
3Q8
3Q8
C12
2Dic3
2Dic3
3Q16
Dic6
C24
Dic6
Dic12

Character table of Dic12

 class 1234A4B4C68A8B12A12B24A24B24C24D
 size 11221212222222222
ρ1111111111111111    trivial
ρ211111-11-1-111-1-1-1-1    linear of order 2
ρ31111-1-1111111111    linear of order 2
ρ41111-111-1-111-1-1-1-1    linear of order 2
ρ5222-200200-2-20000    orthogonal lifted from D4
ρ622-1200-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ722-1200-1-2-2-1-11111    orthogonal lifted from D6
ρ822-1-200-10011-√3√3√3-√3    orthogonal lifted from D12
ρ922-1-200-10011√3-√3-√3√3    orthogonal lifted from D12
ρ102-22000-2-√2√200√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ112-22000-2√2-√200-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ122-2-10001√2-√2√3-√3ζ83ζ3+ζ8ζ3+ζ8ζ87ζ3+ζ87+ζ85ζ3ζ87ζ32+ζ85ζ32+ζ85ζ83ζ32+ζ83+ζ8ζ32    symplectic faithful, Schur index 2
ρ132-2-10001-√2√2√3-√3ζ83ζ32+ζ83+ζ8ζ32ζ87ζ32+ζ85ζ32+ζ85ζ87ζ3+ζ87+ζ85ζ3ζ83ζ3+ζ8ζ3+ζ8    symplectic faithful, Schur index 2
ρ142-2-10001-√2√2-√3√3ζ87ζ32+ζ85ζ32+ζ85ζ83ζ32+ζ83+ζ8ζ32ζ83ζ3+ζ8ζ3+ζ8ζ87ζ3+ζ87+ζ85ζ3    symplectic faithful, Schur index 2
ρ152-2-10001√2-√2-√3√3ζ87ζ3+ζ87+ζ85ζ3ζ83ζ3+ζ8ζ3+ζ8ζ83ζ32+ζ83+ζ8ζ32ζ87ζ32+ζ85ζ32+ζ85    symplectic faithful, Schur index 2

Smallest permutation representation of Dic12
►Regular action on 48 points
Generators in S48
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)

(1 47 13 35)(2 46 14 34)(3 45 15 33)(4 44 16 32)(5 43 17 31)(6 42 18 30)(7 41 19 29)(8 40 20 28)(9 39 21 27)(10 38 22 26)(11 37 23 25)(12 36 24 48)
 
G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,47,13,35)(2,46,14,34)(3,45,15,33)(4,44,16,32)(5,43,17,31)(6,42,18,30)(7,41,19,29)(8,40,20,28)(9,39,21,27)(10,38,22,26)(11,37,23,25)(12,36,24,48)>;
 
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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,47,13,35)(2,46,14,34)(3,45,15,33)(4,44,16,32)(5,43,17,31)(6,42,18,30)(7,41,19,29)(8,40,20,28)(9,39,21,27)(10,38,22,26)(11,37,23,25)(12,36,24,48) );
 
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),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,47,13,35),(2,46,14,34),(3,45,15,33),(4,44,16,32),(5,43,17,31),(6,42,18,30),(7,41,19,29),(8,40,20,28),(9,39,21,27),(10,38,22,26),(11,37,23,25),(12,36,24,48)]])
 

Dic12 is a maximal subgroup of
 C48⋊C2  Dic24  D8.S3  C3⋊Q32  C4○D24  C8.D6  D8⋊3S3  D4.D6  S3×Q16  Dic36  C32⋊3Q16  C32⋊5Q16  A4⋊Q16  C8.S4  C5⋊Dic12  Dic60  C7⋊Dic12  Dic84  He3⋊Q16  C33⋊3Q16
Dic12 is a maximal quotient of
 C2.Dic12  C24⋊1C4  Dic36  C32⋊3Q16  C32⋊5Q16  A4⋊Q16  C5⋊Dic12  Dic60  C7⋊Dic12  Dic84  C33⋊3Q16

Matrix representation of Dic12 ►in GL2(𝔽23) generated by

917
1722
,
022
10
G:=sub<GL(2,GF(23))| [9,17,17,22],[0,1,22,0] >;
 

Dic12 in GAP, Magma, Sage, TeX 

{\rm Dic}_{12}
 
% in TeX
 
G:=Group("Dic12");
 
// GroupNames label
 
G:=SmallGroup(48,8);
 
// by ID
 
G=gap.SmallGroup(48,8);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-3,40,61,66,182,42,804]);
 
// Polycyclic
 
G:=Group<a,b|a^24=1,b^2=a^12,b*a*b^-1=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of Dic12 in TeX
Character table of Dic12 in TeX

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