C2^3.6D6 - GroupNames

G = C₂³.₆D₆ order 96 = 2⁵·3

1^st non-split extension by C₂³ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₆D₆, C₂².₂D₁₂, (C₂²×S₃)⋊C₄, (C₂×Dic₃)⋊C₄, C₂²⋊C₄⋊₁S₃, C₃⋊₁(C₂³⋊C₄), (C₂×C₆).₂₇D₄, C₂.₄(D₆⋊C₄), C₂².₃(C₄×S₃), C₆.D₄⋊₁C₂, C₆.₂(C₂²⋊C₄), C₂².₈(C₃⋊D₄), (C₂²×C₆).₅C₂², (C₂×C₆).₁(C₂×C₄), (C₃×C₂²⋊C₄)⋊₁C₂, (C₂×C₃⋊D₄).₁C₂, SmallGroup(96,13)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂³.₆D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂×C₃⋊D₄ — C₂³.₆D₆

Lower central

C₃ — C₆ — C₂×C₆ — C₂³.₆D₆

Upper central

C₁ — C₂ — C₂³ — C₂²⋊C₄

Generators and relations for C₂³.₆D₆
G = < a,b,c,d,e | a²=b²=c²=1, d⁶=a, e²=abc, ab=ba, eae^-1=ac=ca, ad=da, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=bcd⁵ >

C₁

C₂

₂C₂

₁₂C₂

C₃

C₂²

₄C₄

₄C₂²

₆C₂²

₆C₄

₁₂C₄

₁₂C₂²

C₆

₂C₆

₄S₃

C₂³

₂C₂×C₄

₃C₂³

₃C₂×C₄

₆C₂×C₄

₆D₄

C₂×C₆

₂Dic₃

₂D₆

₄Dic₃

₄D₆

₄C₂×C₆

₄C₁₂

C₂²⋊C₄

₃C₂²⋊C₄

₃C₂×D₄

C₂²×S₃

C₂²×C₆

C₂×Dic₃

₂C₂×C₁₂

₂C₃⋊D₄

₂C₂×Dic₃

₃C₂³⋊C₄

C₂×C₃⋊D₄

C₆.D₄

C₃×C₂²⋊C₄

C₂³.₆D₆

Character table of C₂³.₆D₆

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	12A	12B	12C	12D
size	1	1	2	2	2	12	2	4	4	12	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	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	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	linear of order 2
ρ₅	1	1	-1	1	-1	-1	1	-i	i	i	-i	1	1	-1	-1	1	-1	-i	-i	i	i	linear of order 4
ρ₆	1	1	-1	1	-1	1	1	-i	i	-i	i	-1	1	-1	-1	1	-1	-i	-i	i	i	linear of order 4
ρ₇	1	1	-1	1	-1	-1	1	i	-i	-i	i	1	1	-1	-1	1	-1	i	i	-i	-i	linear of order 4
ρ₈	1	1	-1	1	-1	1	1	i	-i	i	-i	-1	1	-1	-1	1	-1	i	i	-i	-i	linear of order 4
ρ₉	2	2	2	-2	-2	0	2	0	0	0	0	0	2	2	2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	-2	2	0	2	0	0	0	0	0	2	-2	-2	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	0	-1	-2	-2	0	0	0	-1	-1	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₂	2	2	2	2	2	0	-1	2	2	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₃	2	2	-2	-2	2	0	-1	0	0	0	0	0	-1	1	1	1	-1	-√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₁₄	2	2	-2	-2	2	0	-1	0	0	0	0	0	-1	1	1	1	-1	√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₅	2	2	-2	2	-2	0	-1	-2i	2i	0	0	0	-1	1	1	-1	1	i	i	-i	-i	complex lifted from C₄×S₃
ρ₁₆	2	2	-2	2	-2	0	-1	2i	-2i	0	0	0	-1	1	1	-1	1	-i	-i	i	i	complex lifted from C₄×S₃
ρ₁₇	2	2	2	-2	-2	0	-1	0	0	0	0	0	-1	-1	-1	1	1	-√-3	√-3	√-3	-√-3	complex lifted from C₃⋊D₄
ρ₁₈	2	2	2	-2	-2	0	-1	0	0	0	0	0	-1	-1	-1	1	1	√-3	-√-3	-√-3	√-3	complex lifted from C₃⋊D₄
ρ₁₉	4	-4	0	0	0	0	4	0	0	0	0	0	-4	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₀	4	-4	0	0	0	0	-2	0	0	0	0	0	2	-2√-3	2√-3	0	0	0	0	0	0	complex faithful
ρ₂₁	4	-4	0	0	0	0	-2	0	0	0	0	0	2	2√-3	-2√-3	0	0	0	0	0	0	complex faithful

Permutation representations of C₂³.₆D₆
►On 24 points - transitive group 24T94

Generators in S₂₄

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(1 9)(3 11)(5 7)(13 19)(15 21)(17 23)

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

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

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

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

G:=TransitiveGroup(24,94);

►On 24 points - transitive group 24T103

Generators in S₂₄

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

(2 22)(4 24)(6 14)(8 16)(10 18)(12 20)

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

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

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

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

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

G:=TransitiveGroup(24,103);

►On 24 points - transitive group 24T108

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,108);

►On 24 points - transitive group 24T119

Generators in S₂₄

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

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

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

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

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

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

G:=TransitiveGroup(24,119);

C₂³.₆D₆ is a maximal subgroup of
C₂³⋊C₄⋊₅S₃ C₂³⋊D₁₂ C₂³.₅D₁₂ S₃×C₂³⋊C₄ (C₂×D₁₂)⋊₁₃C₄ C₂⁴⋊₆D₆ C₂²⋊C₄⋊D₆ C₂².D₃₆ C₆².₃₁D₄ C₆².₃₂D₄ C₆².₁₁₀D₄ C₁₅⋊₈(C₂³⋊C₄) C₁₅⋊₉(C₂³⋊C₄) C₂³.₆D₃₀ D₁₀.D₁₂ D₁₀.₄D₁₂
C₂³.₆D₆ is a maximal quotient of
C₆.C₄≀C₂ C₄⋊Dic₃⋊C₄ C₂³.₃₅D₁₂ (C₂²×S₃)⋊C₈ (C₂×Dic₃)⋊C₈ C₂².₂D₂₄ C₃⋊C₂≀C₄ (C₂×D₄).D₆ C₂³.D₁₂ C₂³.₂D₁₂ C₂³.₃D₁₂ C₂³.₄D₁₂ (C₂×C₄).D₁₂ (C₂×C₁₂).D₄ C₂⁴.₁₃D₆ C₂².D₃₆ C₆².₃₁D₄ C₆².₃₂D₄ C₆².₁₁₀D₄ C₁₅⋊₈(C₂³⋊C₄) C₁₅⋊₉(C₂³⋊C₄) C₂³.₆D₃₀ D₁₀.D₁₂ D₁₀.₄D₁₂

Matrix representation of C₂³.₆D₆ ►in GL₄(𝔽₇) generated by

0	1	4	5
1	0	3	5
0	0	1	0
0	0	0	6

1	5	2	6
1	5	5	3
0	0	1	0
5	2	1	0

6	0	0	0
0	6	0	0
0	0	6	0
0	0	0	6

1	0	5	2
2	1	6	1
5	5	5	4
4	3	5	0

4	2	2	6
3	0	5	4
6	1	4	3
4	4	5	6

G:=sub<GL(4,GF(7))| [0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[1,1,0,5,5,5,0,2,2,5,1,1,6,3,0,0],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,2,5,4,0,1,5,3,5,6,5,5,2,1,4,0],[4,3,6,4,2,0,1,4,2,5,4,5,6,4,3,6] >;

C₂³.₆D₆ in GAP, Magma, Sage, TeX

C_2^3._6D_6

% in TeX

G:=Group("C2^3.6D6");

// GroupNames label

G:=SmallGroup(96,13);

// by ID

G=gap.SmallGroup(96,13);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,297,2309]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=a,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^5>;

// generators/relations

Export

Subgroup lattice of C₂³.₆D₆ in TeX
Character table of C₂³.₆D₆ in TeX

G = C23.6D6 order 96 = 25·3

1st non-split extension by C23 of D6 acting via D6/C3=C22

G = C₂³.₆D₆ order 96 = 2⁵·3

1^st non-split extension by C₂³ of D₆ acting via D₆/C₃=C₂²