C8.D6 - GroupNames

G = C₈.D₆ order 96 = 2⁵·3

1^st non-split extension by C₈ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₈.₁D₆, C₄.₁₅D₁₂, C₁₂.₁₃D₄, Dic₁₂⋊₂C₂, M₄(2)⋊₂S₃, C₂₄.₁C₂², C₂².₆D₁₂, C₁₂.₃₃C₂³, D₁₂.₈C₂², Dic₆.₈C₂², (C₂×C₆).₆D₄, C₂₄⋊C₂⋊₂C₂, C₆.₁₄(C₂×D₄), (C₂×C₄).₁₆D₆, (C₂×Dic₆)⋊₈C₂, C₄○D₁₂.₄C₂, C₂.₁₆(C₂×D₁₂), C₃⋊₁(C₈.C₂²), (C₃×M₄(2))⋊₂C₂, C₄.₃₁(C₂²×S₃), (C₂×C₁₂).₂₈C₂², SmallGroup(96,116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₈.D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — D₁₂ — C₄○D₁₂ — C₈.D₆

Lower central

C₃ — C₆ — C₁₂ — C₈.D₆

Upper central

C₁ — C₂ — C₂×C₄ — M₄(2)

Generators and relations for C₈.D₆
G = < a,b,c | a⁸=1, b⁶=c²=a⁴, bab^-1=a⁵, cac^-1=a^-1, cbc^-1=b⁵ >

Subgroups: 146 in 60 conjugacy classes, 29 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Dic₃, C₁₂, D₆, C₂×C₆, M₄(2), SD₁₆, Q₁₆, C₂×Q₈, C₄○D₄, C₂₄, Dic₆, Dic₆, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₈.C₂², C₂₄⋊C₂, Dic₁₂, C₃×M₄(2), C₂×Dic₆, C₄○D₁₂, C₈.D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, D₁₂, C₂²×S₃, C₈.C₂², C₂×D₁₂, C₈.D₆

Character table of C₈.D₆

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	6A	6B	8A	8B	12A	12B	12C	24A	24B	24C	24D
size	1	1	2	12	2	2	2	12	12	12	2	4	4	4	2	2	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	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	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	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	linear of order 2
ρ₉	2	2	2	0	-1	2	2	0	0	0	-1	-1	-2	-2	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₀	2	2	2	0	-1	2	2	0	0	0	-1	-1	2	2	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₁	2	2	2	0	2	-2	-2	0	0	0	2	2	0	0	-2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	0	-1	-2	2	0	0	0	-1	1	2	-2	-1	-1	1	1	-1	-1	1	orthogonal lifted from D₆
ρ₁₃	2	2	-2	0	-1	-2	2	0	0	0	-1	1	-2	2	-1	-1	1	-1	1	1	-1	orthogonal lifted from D₆
ρ₁₄	2	2	-2	0	2	2	-2	0	0	0	2	-2	0	0	-2	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	0	-1	-2	-2	0	0	0	-1	-1	0	0	1	1	1	-√3	-√3	√3	√3	orthogonal lifted from D₁₂
ρ₁₆	2	2	2	0	-1	-2	-2	0	0	0	-1	-1	0	0	1	1	1	√3	√3	-√3	-√3	orthogonal lifted from D₁₂
ρ₁₇	2	2	-2	0	-1	2	-2	0	0	0	-1	1	0	0	1	1	-1	-√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₁₈	2	2	-2	0	-1	2	-2	0	0	0	-1	1	0	0	1	1	-1	√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₉	4	-4	0	0	4	0	0	0	0	0	-4	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₀	4	-4	0	0	-2	0	0	0	0	0	2	0	0	0	-2√3	2√3	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₂₁	4	-4	0	0	-2	0	0	0	0	0	2	0	0	0	2√3	-2√3	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₈.D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

C₈.D₆ is a maximal subgroup of
M₄(2)⋊D₆ D₁₂.₂D₄ D₁₂.₄D₄ D₁₂.₇D₄ C₄²⋊₅D₆ Q₈.₁₄D₁₂ C₂₄.₁₈D₄ C₂₄.₄₂D₄ C₂₄.₉C₂³ D₄.₁₁D₁₂ D₄.₁₃D₁₂ D₈⋊₄D₆ D₈⋊₆D₆ S₃×C₈.C₂² SD₁₆.D₆ C₈.D₁₈ C₂₄.₃D₆ Dic₁₂⋊S₃ D₁₂.₂₉D₆ Dic₆.₂₉D₆ C₂₄.₅D₆ Dic₆₀⋊C₂ C₂₄.₂D₁₀ C₂₀.D₁₂ D₁₂.₃₃D₁₀ C₈.D₃₀
C₈.D₆ is a maximal quotient of
C₈⋊Dic₆ C₄².₁₄D₆ C₄².₁₆D₆ C₄².₂₀D₆ C₈.D₁₂ Dic₁₂⋊C₄ C₂³.₃₉D₁₂ C₂³.₁₅D₁₂ D₁₂.₃₂D₄ C₂².D₂₄ Dic₆⋊₁₄D₄ Dic₆.₃₂D₄ Dic₆.₃Q₈ C₁₂⋊SD₁₆ C₄².₃₆D₆ D₁₂⋊₄Q₈ C₄⋊Dic₁₂ Dic₆⋊₄Q₈ C₂³.₅₁D₁₂ C₂³.₅₂D₁₂ C₂³.₅₄D₁₂ C₂₄⋊₂D₄ C₂₄.₄D₄ C₈.D₁₈ C₂₄.₃D₆ Dic₁₂⋊S₃ D₁₂.₂₉D₆ Dic₆.₂₉D₆ C₂₄.₅D₆ Dic₆₀⋊C₂ C₂₄.₂D₁₀ C₂₀.D₁₂ D₁₂.₃₃D₁₀ C₈.D₃₀

Matrix representation of C₈.D₆ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	1	1	72	71
0	0	0	1	0	0
0	0	1	0	0	72

0	1	0	0	0	0
72	1	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	27	0
0	0	46	46	0	27

72	1	0	0	0	0
0	1	0	0	0	0
0	0	46	0	0	0
0	0	0	27	0	0
0	0	27	27	46	19
0	0	46	0	0	27

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,1,1,0,0,0,1,72,0,0,0,0,0,71,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,46,0,0,46,0,0,0,46,0,46,0,0,0,0,27,0,0,0,0,0,0,27],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,46,0,27,46,0,0,0,27,27,0,0,0,0,0,46,0,0,0,0,0,19,27] >;

C₈.D₆ in GAP, Magma, Sage, TeX

C_8.D_6

% in TeX

G:=Group("C8.D6");

// GroupNames label

G:=SmallGroup(96,116);

// by ID

G=gap.SmallGroup(96,116);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,50,579,69,2309]);

// Polycyclic

G:=Group<a,b,c|a^8=1,b^6=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^5>;

// generators/relations

Export

Character table of C₈.D₆ in TeX

G = C8.D6 order 96 = 25·3

1st non-split extension by C8 of D6 acting via D6/C3=C22

G = C₈.D₆ order 96 = 2⁵·3

1^st non-split extension by C₈ of D₆ acting via D₆/C₃=C₂²