C8:6D4 - GroupNames

G = C₈:₆D₄ order 64 = 2⁶

3^rd semidirect product of C₈ and D₄ acting via D₄/C₄=C₂

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₈:₆D₄, C₄:₁M₄(2), C₄².₇₀C₂², C₄:C₈:₁₆C₂, (C₄xC₈):₁₅C₂, C₄:C₄.₈C₄, (C₂xD₄).₉C₄, (C₄xD₄).₃C₂, C₂.₁₁(C₄xD₄), C₄.₇₈(C₂xD₄), C₂²:C₈:₁₄C₂, C₂.₈(C₈oD₄), C₂²:C₄.₅C₄, C₄.₅₃(C₄oD₄), C₂³.₁₂(C₂xC₄), (C₂xM₄(2)):₁₅C₂, (C₂xC₄).₁₅₅C₂³, (C₂xC₈).₁₀₁C₂², C₂.₁₀(C₂xM₄(2)), C₂².₄₈(C₂²xC₄), (C₂²xC₄).₄₁C₂², (C₂xC₄).₂₉(C₂xC₄), SmallGroup(64,117)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂² — C₈:₆D₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₈ — C₂xM₄(2) — C₈:₆D₄

Lower central

C₁ — C₂² — C₈:₆D₄

Upper central

C₁ — C₂xC₄ — C₈:₆D₄

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₈:₆D₄

Generators and relations for C₈:₆D₄
G = < a,b,c | a⁸=b⁴=c²=1, ab=ba, cac=a⁵, cbc=b^-1 >

Subgroups: 89 in 61 conjugacy classes, 37 normal (19 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₈, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, C₂³, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, M₄(2), C₂²xC₄, C₂xD₄, C₄xC₈, C₂²:C₈, C₄:C₈, C₄xD₄, C₂xM₄(2), C₈:₆D₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, M₄(2), C₂²xC₄, C₂xD₄, C₄oD₄, C₄xD₄, C₂xM₄(2), C₈oD₄, C₈:₆D₄

Character table of C₈:₆D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L
size	1	1	1	1	4	4	1	1	1	1	2	2	2	2	4	4	2	2	2	2	2	2	2	2	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	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	-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	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	-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	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	-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	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	-1	1	-1	1	linear of order 2
ρ₉	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	-1	1	-1	1	-i	i	i	i	-i	i	-i	-i	i	-i	-i	i	linear of order 4
ρ₁₀	1	1	1	1	1	-1	-1	-1	-1	-1	1	-1	-1	1	1	-1	-i	i	i	i	-i	i	-i	-i	-i	i	i	-i	linear of order 4
ρ₁₁	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	-1	1	-1	1	i	-i	-i	-i	i	-i	i	i	-i	i	i	-i	linear of order 4
ρ₁₂	1	1	1	1	1	-1	-1	-1	-1	-1	1	-1	-1	1	1	-1	i	-i	-i	-i	i	-i	i	i	i	-i	-i	i	linear of order 4
ρ₁₃	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	1	1	-1	1	1	-i	-i	i	-i	-i	i	i	i	i	i	-i	-i	linear of order 4
ρ₁₄	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-1	-1	-1	-i	-i	i	-i	-i	i	i	i	-i	-i	i	i	linear of order 4
ρ₁₅	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	1	1	-1	1	1	i	i	-i	i	i	-i	-i	-i	-i	-i	i	i	linear of order 4
ρ₁₆	1	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-1	-1	-1	i	i	-i	i	i	-i	-i	-i	i	i	-i	-i	linear of order 4
ρ₁₇	2	-2	2	-2	0	0	-2	2	-2	2	0	0	0	0	0	0	2	0	-2	0	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	2	-2	0	0	-2	2	-2	2	0	0	0	0	0	0	-2	0	2	0	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	-2	-2	0	0	2i	-2i	-2i	2i	-2i	2	-2	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₀	2	2	-2	-2	0	0	-2i	2i	2i	-2i	2i	2	-2	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₁	2	2	-2	-2	0	0	-2i	2i	2i	-2i	-2i	-2	2	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₂	2	2	-2	-2	0	0	2i	-2i	-2i	2i	2i	-2	2	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₃	2	-2	2	-2	0	0	2	-2	2	-2	0	0	0	0	0	0	2i	0	2i	0	-2i	-2i	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₄	2	-2	2	-2	0	0	2	-2	2	-2	0	0	0	0	0	0	-2i	0	-2i	0	2i	2i	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₅	2	-2	-2	2	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	2ζ₈⁵	0	2ζ₈	0	0	2ζ₈⁷	2ζ₈³	0	0	0	0	complex lifted from C₈oD₄
ρ₂₆	2	-2	-2	2	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	2ζ₈³	0	2ζ₈⁷	0	0	2ζ₈	2ζ₈⁵	0	0	0	0	complex lifted from C₈oD₄
ρ₂₇	2	-2	-2	2	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	2ζ₈	0	2ζ₈⁵	0	0	2ζ₈³	2ζ₈⁷	0	0	0	0	complex lifted from C₈oD₄
ρ₂₈	2	-2	-2	2	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	2ζ₈⁷	0	2ζ₈³	0	0	2ζ₈⁵	2ζ₈	0	0	0	0	complex lifted from C₈oD₄

Smallest permutation representation of C₈:₆D₄
►On 32 points

Generators in S₃₂

(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)

(1 14 18 25)(2 15 19 26)(3 16 20 27)(4 9 21 28)(5 10 22 29)(6 11 23 30)(7 12 24 31)(8 13 17 32)

(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)

G:=sub<Sym(32)| (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), (1,14,18,25)(2,15,19,26)(3,16,20,27)(4,9,21,28)(5,10,22,29)(6,11,23,30)(7,12,24,31)(8,13,17,32), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(16,20)>;

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), (1,14,18,25)(2,15,19,26)(3,16,20,27)(4,9,21,28)(5,10,22,29)(6,11,23,30)(7,12,24,31)(8,13,17,32), (1,25)(2,30)(3,27)(4,32)(5,29)(6,26)(7,31)(8,28)(9,17)(10,22)(11,19)(12,24)(13,21)(14,18)(15,23)(16,20) );

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)], [(1,14,18,25),(2,15,19,26),(3,16,20,27),(4,9,21,28),(5,10,22,29),(6,11,23,30),(7,12,24,31),(8,13,17,32)], [(1,25),(2,30),(3,27),(4,32),(5,29),(6,26),(7,31),(8,28),(9,17),(10,22),(11,19),(12,24),(13,21),(14,18),(15,23),(16,20)]])

C₈:₆D₄ is a maximal subgroup of
C₈.₃₁D₈ C₈:₁₅SD₁₆ Q₈:₂M₄(2) C₈.D₈ C₈:₃SD₁₆ C₈:₄SD₁₆ C₈.₈SD₁₆ C₄².₂₄₈C₂³ C₄².₂₅₀C₂³ C₄².₂₅₂C₂³ C₄².₂₅₄C₂³ C₄².₂₆₅C₂³ C₄².₆₈₁C₂³ M₄(2):₂₂D₄ D₄xM₄(2) M₄(2):₂₃D₄ C₄².₂₉₀C₂³ C₄².₂₉₁C₂³ C₄².₂₉₂C₂³ C₄².₂₉₄C₂³ Q₈:₆M₄(2) D₄:₇M₄(2) C₄².₆₉₃C₂³ C₄².₂₉₇C₂³ C₄².₂₉₉C₂³ C₄².₆₉₄C₂³ C₄².₃₀₁C₂³ C₄².₆₉₈C₂³ Q₈:₇M₄(2) C₄².₃₀₈C₂³ C₄².₃₀₉C₂³ C₄².₃₁₀C₂³ D₈:₄D₄ D₈:₅D₄ SD₁₆:₁D₄ SD₁₆:₂D₄ SD₁₆:₃D₄ Q₁₆:₄D₄ Q₁₆:₅D₄ C₄².₄₇₁C₂³ C₄².₄₇₂C₂³ C₄².₄₇₃C₂³ C₄².₄₇₄C₂³ C₄².₄₇₅C₂³ C₄².₄₇₆C₂³ C₄².₄₇₇C₂³ C₄².₄₇₈C₂³ C₄².₄₇₉C₂³ C₄².₄₈₀C₂³ C₄².₄₈₁C₂³ C₄².₄₈₂C₂³ C₄².₄₉₂C₂³ C₄².₄₉₃C₂³ C₄².₄₉₄C₂³ C₄².₄₉₅C₂³ C₄².₄₉₆C₂³ C₄².₄₉₇C₂³ C₄².₄₉₈C₂³ Dic₅:₂M₄(2) C₄₀:₁₈D₄
C₈:D_4p: C₈:₉D₈ C₈:₆D₈ C₈:D₈ C₈:₂D₈ C₈:₆D₁₂ C₈:₆D₂₀ C₈:₆D₂₈ ...
C_4p:M₄(2): C₈:₉SD₁₆ C₈:M₄(2) C₈:₃M₄(2) C₁₂:₂M₄(2) C₁₂:₃M₄(2) C₂₀:₆M₄(2) C₂₀:₇M₄(2) C₂₀:M₄(2) ...
D_2p:C₈:C₂: D₄:₂M₄(2) Dic₃:M₄(2) C₂₄:₂₁D₄ Dic₅:M₄(2) Dic₇:M₄(2) C₅₆:₁₈D₄ ...
C₈:₆D₄ is a maximal quotient of
C₈:₆Q₁₆ C₈.M₄(2) (C₂xC₈).₁₉₅D₄ C₂³:₂M₄(2) (C₂xC₈).Q₈ C₂²:C₄:₄C₈ C₂³.₉M₄(2) C₄².₆₁Q₈ C₄².₃₂₅D₄
C₈:D_4p: C₈:₆D₈ C₈:₆D₁₂ C₈:₆D₂₀ C₈:₆D₂₈ ...
C_4p:M₄(2): C₈:₉SD₁₆ C₈:M₄(2) C₈:₃M₄(2) C₁₂:₂M₄(2) C₁₂:₃M₄(2) C₂₀:₆M₄(2) C₂₀:₇M₄(2) C₂₀:M₄(2) ...
C_2p.(C₄xD₄): C₂³.₁₇C₄² C₄:C₈:₁₃C₄ Dic₃:M₄(2) C₂₄:₂₁D₄ Dic₅:₂M₄(2) C₄₀:₁₈D₄ Dic₅:M₄(2) Dic₇:M₄(2) ...

Matrix representation of C₈:₆D₄ ►in GL₄(F₁₇) generated by

5	15	0	0
2	12	0	0
0	0	13	0
0	0	0	13

10	13	0	0
4	7	0	0
0	0	1	8
0	0	4	16

10	13	0	0
12	7	0	0
0	0	1	8
0	0	0	16

G:=sub<GL(4,GF(17))| [5,2,0,0,15,12,0,0,0,0,13,0,0,0,0,13],[10,4,0,0,13,7,0,0,0,0,1,4,0,0,8,16],[10,12,0,0,13,7,0,0,0,0,1,0,0,0,8,16] >;

C₈:₆D₄ in GAP, Magma, Sage, TeX

C_8\rtimes_6D_4

% in TeX

G:=Group("C8:6D4");

// GroupNames label

G:=SmallGroup(64,117);

// by ID

G=gap.SmallGroup(64,117);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,332,86,88]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈:₆D₄ in TeX

G = C8:6D4 order 64 = 26

3rd semidirect product of C8 and D4 acting via D4/C4=C2

G = C₈:₆D₄ order 64 = 2⁶

3^rd semidirect product of C₈ and D₄ acting via D₄/C₄=C₂