C2^3:2D6 - GroupNames

G = C₂³:₂D₆ order 96 = 2⁵·3

1^st semidirect product of C₂³ and D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₆:₅D₄, C₂³:₂D₆, (C₂xC₄):₂D₆, (C₂xC₆):₂D₄, C₃:₂C₂²wrC₂, (C₂xD₄):₃S₃, (C₆xD₄):₈C₂, D₆:C₄:₁₄C₂, C₂.₂₅(S₃xD₄), C₆.₄₉(C₂xD₄), (S₃xC₂³):₂C₂, (C₂xC₁₂):₇C₂², C₂²:₃(C₃:D₄), (C₂xC₆).₅₂C₂³, (C₂²xC₆):₃C₂², C₆.D₄:₁₀C₂, (C₂xDic₃):₂C₂², C₂².₅₉(C₂²xS₃), (C₂²xS₃).₂₅C₂², (C₂xC₃:D₄):₄C₂, C₂.₁₃(C₂xC₃:D₄), SmallGroup(96,144)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂³:₂D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — C₂³:₂D₆

Lower central

C₃ — C₂xC₆ — C₂³:₂D₆

Upper central

C₁ — C₂² — C₂xD₄

Generators and relations for C₂³:₂D₆
G = < a,b,c,d,e | a²=b²=c²=d⁶=e²=1, ab=ba, dad^-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 354 in 130 conjugacy classes, 37 normal (17 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₂xD₄, C₂xD₄, C₂⁴, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²wrC₂, D₆:C₄, C₆.D₄, C₂xC₃:D₄, C₆xD₄, S₃xC₂³, C₂³:₂D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₂²wrC₂, S₃xD₄, C₂xC₃:D₄, C₂³:₂D₆

Character table of C₂³:₂D₆

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	3	4A	4B	4C	6A	6B	6C	6D	6E	6F	6G	12A	12B
size	1	1	1	1	2	2	4	6	6	6	6	2	4	12	12	2	2	2	4	4	4	4	4	4
ρ₁	1	1	1	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	-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	-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	1	1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	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
ρ₆	1	1	1	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
ρ₇	1	1	1	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
ρ₈	1	1	1	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
ρ₉	2	2	2	2	2	2	-2	0	0	0	0	-1	-2	0	0	-1	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	-2	-2	2	0	0	0	-2	0	0	2	2	0	0	0	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	0	0	2	0	0	-2	2	0	0	0	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	2	-2	0	0	0	0	2	-2	0	2	0	0	0	2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	2	0	0	0	0	-1	-2	0	0	-1	-1	-1	1	1	-1	-1	1	1	orthogonal lifted from D₆
ρ₁₄	2	-2	2	-2	0	0	0	0	-2	2	0	2	0	0	0	2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	2	-2	-2	-2	0	0	0	0	-1	2	0	0	-1	-1	-1	1	1	1	1	-1	-1	orthogonal lifted from D₆
ρ₁₆	2	2	-2	-2	2	-2	0	0	0	0	0	2	0	0	0	-2	2	-2	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	2	2	2	2	2	2	0	0	0	0	-1	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₈	2	2	-2	-2	-2	2	0	0	0	0	0	2	0	0	0	-2	2	-2	2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	-2	-2	-2	2	0	0	0	0	0	-1	0	0	0	1	-1	1	-1	1	√-3	-√-3	√-3	-√-3	complex lifted from C₃:D₄
ρ₂₀	2	2	-2	-2	2	-2	0	0	0	0	0	-1	0	0	0	1	-1	1	1	-1	-√-3	√-3	√-3	-√-3	complex lifted from C₃:D₄
ρ₂₁	2	2	-2	-2	2	-2	0	0	0	0	0	-1	0	0	0	1	-1	1	1	-1	√-3	-√-3	-√-3	√-3	complex lifted from C₃:D₄
ρ₂₂	2	2	-2	-2	-2	2	0	0	0	0	0	-1	0	0	0	1	-1	1	-1	1	-√-3	√-3	-√-3	√-3	complex lifted from C₃:D₄
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	-2	0	0	0	-2	2	2	0	0	0	0	0	0	orthogonal lifted from S₃xD₄
ρ₂₄	4	-4	-4	4	0	0	0	0	0	0	0	-2	0	0	0	2	2	-2	0	0	0	0	0	0	orthogonal lifted from S₃xD₄

Permutation representations of C₂³:₂D₆
►On 24 points - transitive group 24T145

Generators in S₂₄

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

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

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

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

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

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

G:=TransitiveGroup(24,145);

C₂³:₂D₆ is a maximal subgroup of
C₃:C₂wrC₄ C₂³.₃D₁₂ C₂³:D₁₂ 2₊¹⁺⁴:₇S₃ C₄²:₁₄D₆ D₁₂:₂₃D₄ C₄²:₁₈D₆ C₄²:₁₉D₆ S₃xC₂²wrC₂ C₂⁴:₇D₆ C₂⁴:₈D₆ C₂⁴.₄₄D₆ C₂⁴.₄₅D₆ C₂⁴:₉D₆ C₆.₃₇2₊¹⁺⁴ C₄:C₄:₂₁D₆ C₆.₃₈2₊¹⁺⁴ D₁₂:₁₉D₄ C₆.₄₀2₊¹⁺⁴ D₁₂:₂₀D₄ C₆.₄₂2₊¹⁺⁴ C₆.₄₆2₊¹⁺⁴ C₆.₄₈2₊¹⁺⁴ C₆.₁₂₀2₊¹⁺⁴ C₄:C₄:₂₈D₆ C₆.₆₁2₊¹⁺⁴ C₆.₁₂₂2₊¹⁺⁴ C₆.₆₂2₊¹⁺⁴ C₆.₆₈2₊¹⁺⁴ C₄²:₂₂D₆ C₄²:₂₃D₆ C₄²:₂₄D₆ C₄²:₂₈D₆ D₁₂:₁₁D₄ C₄²:₃₀D₆ D₄xC₃:D₄ C₂⁴:₁₂D₆ (C₂xD₄):₄₃D₆ C₆.₁₄₅2₊¹⁺⁴ C₆.₁₄₆2₊¹⁺⁴ C₂³:₂D₁₈ D₆:₄D₁₂ C₆²:₄D₄ C₆²:₈D₄ C₆².₁₂₅C₂³ C₆²:₁₃D₄ D₆:S₄ D₆:₄D₂₀ C₁₅:C₂²wrC₂ D₃₀:₁₈D₄ D₃₀:₈D₄ D₃₀:₁₇D₄
C₂³:₂D₆ is a maximal quotient of
C₂⁴.₅₇D₆ C₂³:₂Dic₆ C₂⁴.₅₉D₆ C₂⁴.₂₃D₆ C₂⁴.₂₅D₆ C₂³:₃D₁₂ (C₂xDic₃):Q₈ D₆:C₄:₆C₄ (C₂xC₄):₃D₁₂ C₂⁴:₆D₆ D₁₂:₁₆D₄ D₁₂:₁₇D₄ Dic₆:₁₇D₄ D₁₂.₃₆D₄ D₁₂.₃₇D₄ Dic₆.₃₇D₄ C₂²:C₄:D₆ C₄²:₇D₆ D₁₂.₁₄D₄ C₄²:₈D₆ D₁₂.₁₅D₄ D₁₂:D₄ Dic₆:D₄ D₆:₆SD₁₆ D₆:₈SD₁₆ D₁₂:₇D₄ Dic₆.₁₆D₄ D₆:₅Q₁₆ D₁₂.₁₇D₄ D₁₂:₁₈D₄ D₁₂.₃₈D₄ D₁₂.₃₉D₄ D₁₂.₄₀D₄ C₂⁴.₂₉D₆ C₂⁴.₃₂D₆ C₂³:₂D₁₈ D₆:₄D₁₂ C₆²:₄D₄ C₆²:₈D₄ C₆².₁₂₅C₂³ C₆²:₁₃D₄ D₆:₄D₂₀ C₁₅:C₂²wrC₂ D₃₀:₁₈D₄ D₃₀:₈D₄ D₃₀:₁₇D₄

Matrix representation of C₂³:₂D₆ ►in GL₄(F₁₃) generated by

2	4	0	0
9	11	0	0
0	0	0	3
0	0	9	0

12	0	0	0
0	12	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	1	0	0
0	0	12	0
0	0	0	12

1	1	0	0
12	0	0	0
0	0	12	0
0	0	0	1

1	1	0	0
0	12	0	0
0	0	1	0
0	0	0	12

G:=sub<GL(4,GF(13))| [2,9,0,0,4,11,0,0,0,0,0,9,0,0,3,0],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,12,0,0,1,0,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,1,12,0,0,0,0,1,0,0,0,0,12] >;

C₂³:₂D₆ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_6

% in TeX

G:=Group("C2^3:2D6");

// GroupNames label

G:=SmallGroup(96,144);

// by ID

G=gap.SmallGroup(96,144);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,188,2309]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³:₂D₆ in TeX

G = C23:2D6 order 96 = 25·3

1st semidirect product of C23 and D6 acting via D6/C3=C22

G = C₂³:₂D₆ order 96 = 2⁵·3

1^st semidirect product of C₂³ and D₆ acting via D₆/C₃=C₂²