Dic3.A4 - GroupNames

G = Dic₃.A₄ order 144 = 2⁴·3²

The non-split extension by Dic₃ of A₄ acting through Inn(Dic₃)

Aliases: Dic₃.A₄, SL₂(𝔽₃)⋊₂S₃, C₃⋊(C₄.A₄), (C₃×Q₈).C₆, C₂.₂(S₃×A₄), C₆.₁(C₂×A₄), Q₈⋊₃S₃⋊C₃, Q₈.₂(C₃×S₃), (C₃×SL₂(𝔽₃))⋊₂C₂, SmallGroup(144,127)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₃×Q₈ — Dic₃.A₄

Chief series

C₁ — C₂ — C₆ — C₃×Q₈ — C₃×SL₂(𝔽₃) — Dic₃.A₄

Lower central

C₃×Q₈ — Dic₃.A₄

Upper central

C₁ — C₂

Generators and relations for Dic₃.A₄
G = < a,b,c,d,e | a⁶=e³=1, b²=c²=d²=a³, bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=a³c, ece^-1=a³cd, ede^-1=c >

C₁

C₂

₁₈C₂

C₃

₄C₃

₈C₃

₃C₄

₉C₂²

C₆

₄C₆

₆S₃

₈C₆

₄C₃²

Q₈

₉C₂×C₄

₉D₄

Dic₃

₃C₁₂

₃D₆

₁₂C₁₂

₄C₃×C₆

₃C₄○D₄

SL₂(𝔽₃)

C₃×Q₈

₂SL₂(𝔽₃)

₃C₄×S₃

₃D₁₂

₄C₃×Dic₃

Q₈⋊₃S₃

₃C₄.A₄

C₃×SL₂(𝔽₃)

Dic₃.A₄

Character table of Dic₃.A₄

class	1	2A	2B	3A	3B	3C	3D	3E	4A	4B	4C	6A	6B	6C	6D	6E	12A	12B	12C	12D	12E
size	1	1	18	2	4	4	8	8	3	3	6	2	4	4	8	8	12	12	12	12	12
ρ₁	1	1	1	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	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	ζ₃²	ζ₃	ζ₃	linear of order 3
ρ₄	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃	ζ₃	ζ₃²	ζ₃²	linear of order 3
ρ₅	1	1	-1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	-1	-1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	linear of order 6
ρ₆	1	1	-1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	-1	-1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₇	2	2	0	-1	2	2	-1	-1	0	0	2	-1	2	2	-1	-1	-1	0	0	0	0	orthogonal lifted from S₃
ρ₈	2	2	0	-1	-1+√-3	-1-√-3	ζ₆⁵	ζ₆	0	0	2	-1	-1+√-3	-1-√-3	ζ₆⁵	ζ₆	-1	0	0	0	0	complex lifted from C₃×S₃
ρ₉	2	2	0	-1	-1-√-3	-1+√-3	ζ₆	ζ₆⁵	0	0	2	-1	-1-√-3	-1+√-3	ζ₆	ζ₆⁵	-1	0	0	0	0	complex lifted from C₃×S₃
ρ₁₀	2	-2	0	2	-1	-1	-1	-1	-2i	2i	0	-2	1	1	1	1	0	-i	i	-i	i	complex lifted from C₄.A₄
ρ₁₁	2	-2	0	2	-1	-1	-1	-1	2i	-2i	0	-2	1	1	1	1	0	i	-i	i	-i	complex lifted from C₄.A₄
ρ₁₂	2	-2	0	2	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	2i	-2i	0	-2	ζ₃	ζ₃²	ζ₃	ζ₃²	0	ζ₄ζ₃²	ζ₄³ζ₃²	ζ₄ζ₃	ζ₄³ζ₃	complex lifted from C₄.A₄
ρ₁₃	2	-2	0	2	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	-2i	2i	0	-2	ζ₃²	ζ₃	ζ₃²	ζ₃	0	ζ₄³ζ₃	ζ₄ζ₃	ζ₄³ζ₃²	ζ₄ζ₃²	complex lifted from C₄.A₄
ρ₁₄	2	-2	0	2	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	2i	-2i	0	-2	ζ₃²	ζ₃	ζ₃²	ζ₃	0	ζ₄ζ₃	ζ₄³ζ₃	ζ₄ζ₃²	ζ₄³ζ₃²	complex lifted from C₄.A₄
ρ₁₅	2	-2	0	2	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	-2i	2i	0	-2	ζ₃	ζ₃²	ζ₃	ζ₃²	0	ζ₄³ζ₃²	ζ₄ζ₃²	ζ₄³ζ₃	ζ₄ζ₃	complex lifted from C₄.A₄
ρ₁₆	3	3	-1	3	0	0	0	0	3	3	-1	3	0	0	0	0	-1	0	0	0	0	orthogonal lifted from A₄
ρ₁₇	3	3	1	3	0	0	0	0	-3	-3	-1	3	0	0	0	0	-1	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₈	4	-4	0	-2	-2	-2	1	1	0	0	0	2	2	2	-1	-1	0	0	0	0	0	orthogonal faithful, Schur index 2
ρ₁₉	4	-4	0	-2	1-√-3	1+√-3	ζ₃	ζ₃²	0	0	0	2	-1+√-3	-1-√-3	ζ₆⁵	ζ₆	0	0	0	0	0	complex faithful
ρ₂₀	4	-4	0	-2	1+√-3	1-√-3	ζ₃²	ζ₃	0	0	0	2	-1-√-3	-1+√-3	ζ₆	ζ₆⁵	0	0	0	0	0	complex faithful
ρ₂₁	6	6	0	-3	0	0	0	0	0	0	-2	-3	0	0	0	0	1	0	0	0	0	orthogonal lifted from S₃×A₄

Smallest permutation representation of Dic₃.A₄
►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 28 4 25)(2 27 5 30)(3 26 6 29)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)

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

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

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

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

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

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

Dic₃.A₄ is a maximal subgroup of
CSU₂(𝔽₃)⋊S₃ Dic₃.₄S₄ Dic₃.₅S₄ GL₂(𝔽₃)⋊S₃ SL₂(𝔽₃).₁₁D₆ Dic₆.A₄ S₃×C₄.A₄ Dic₉.A₄ Dic₉.₂A₄ C₆.(S₃×A₄) C₃⋊Dic₃.₂A₄
Dic₃.A₄ is a maximal quotient of
Dic₃×SL₂(𝔽₃) Dic₉.A₄ Dic₉.₂A₄ Q₈⋊C₉⋊₃S₃ C₆.(S₃×A₄) C₃⋊Dic₃.₂A₄

Matrix representation of Dic₃.A₄ ►in GL₄(𝔽₅) generated by

0	2	2	1
4	1	4	3
4	2	0	2
3	1	4	1

0	1	3	0
0	0	4	1
3	0	2	3
3	4	2	3

2	4	3	1
1	0	4	2
2	2	2	3
0	1	2	1

4	4	4	4
1	2	4	4
1	1	0	3
0	3	2	4

4	0	1	4
0	4	2	0
0	2	0	0
1	2	0	0

G:=sub<GL(4,GF(5))| [0,4,4,3,2,1,2,1,2,4,0,4,1,3,2,1],[0,0,3,3,1,0,0,4,3,4,2,2,0,1,3,3],[2,1,2,0,4,0,2,1,3,4,2,2,1,2,3,1],[4,1,1,0,4,2,1,3,4,4,0,2,4,4,3,4],[4,0,0,1,0,4,2,2,1,2,0,0,4,0,0,0] >;

Dic₃.A₄ in GAP, Magma, Sage, TeX

{\rm Dic}_3.A_4

% in TeX

G:=Group("Dic3.A4");

// GroupNames label

G:=SmallGroup(144,127);

// by ID

G=gap.SmallGroup(144,127);

# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-2,432,170,230,81,351,165,1444]);

// Polycyclic

G:=Group<a,b,c,d,e|a^6=e^3=1,b^2=c^2=d^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^3*c,e*c*e^-1=a^3*c*d,e*d*e^-1=c>;

// generators/relations

Export

Subgroup lattice of Dic₃.A₄ in TeX
Character table of Dic₃.A₄ in TeX

G = Dic3.A4 order 144 = 24·32

The non-split extension by Dic3 of A4 acting through Inn(Dic3)

G = Dic₃.A₄ order 144 = 2⁴·3²

The non-split extension by Dic₃ of A₄ acting through Inn(Dic₃)