C8:9D4 - GroupNames

G = C₈:₉D₄ order 64 = 2⁶

3^rd semidirect product of C₈ and D₄ acting via D₄/C₂²=C₂

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

Aliases: C₈:₉D₄, C₂²:₁M₄(2), C₄².₈C₂², C₄:C₈:₁₅C₂, C₄:C₄.₇C₄, C₈:C₄:₉C₂, (C₄xD₄).₂C₂, (C₂xD₄).₈C₄, C₂.₁₀(C₄xD₄), C₄.₇₇(C₂xD₄), C₂²:C₈:₁₃C₂, (C₂²xC₈):₁₁C₂, C₂.₇(C₈oD₄), C₂²:C₄.₄C₄, C₄.₅₂(C₄oD₄), C₂³.₁₁(C₂xC₄), C₂.₉(C₂xM₄(2)), (C₂xM₄(2)):₁₄C₂, (C₂xC₈).₁₀₀C₂², (C₂xC₄).₁₅₄C₂³, C₂².₄₇(C₂²xC₄), (C₂²xC₄).₉₆C₂², (C₂xC₄).₃₆(C₂xC₄), SmallGroup(64,116)

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

Derived series

C₁ — C₂² — C₈:₉D₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xC₈ — C₂²xC₈ — C₈:₉D₄

Lower central

C₁ — C₂² — C₈:₉D₄

Upper central

C₁ — C₂xC₄ — C₈:₉D₄

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₈:₉D₄

Generators and relations for C₈:₉D₄
G = < a,b,c | a⁸=b⁴=c²=1, bab^-1=cac=a⁵, cbc=b^-1 >

Subgroups: 89 in 62 conjugacy classes, 37 normal (33 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂xC₄, C₂xC₄, D₄, C₂³, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, C₂xC₈, M₄(2), C₂²xC₄, C₂xD₄, C₈:C₄, C₂²:C₈, C₄:C₈, C₄xD₄, C₂²xC₈, C₂xM₄(2), C₈:₉D₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, M₄(2), C₂²xC₄, C₂xD₄, C₄oD₄, C₄xD₄, C₂xM₄(2), C₈oD₄, C₈:₉D₄

Character table of C₈:₉D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L
size	1	1	1	1	2	2	4	1	1	1	1	2	2	4	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	0	-2	2	2	-2	0	0	0	0	0	2	0	0	0	-2	-2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	-2	2	0	0	0	-2	2	2	-2	0	0	0	0	0	-2	0	0	0	2	2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	-2	2	0	0	0	2	-2	-2	2	0	0	0	0	0	-2i	0	0	0	-2i	2i	2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₀	2	-2	2	-2	-2	2	0	-2i	-2i	2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₁	2	-2	2	-2	2	-2	0	-2i	-2i	2i	2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₂	2	-2	2	-2	-2	2	0	2i	2i	-2i	-2i	-2i	2i	0	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	0	2	-2	-2	2	0	0	0	0	0	2i	0	0	0	2i	-2i	-2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₄	2	-2	2	-2	2	-2	0	2i	2i	-2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₅	2	2	-2	-2	0	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	2ζ₈	2ζ₈³	2ζ₈⁷	0	0	0	2ζ₈⁵	0	0	0	0	complex lifted from C₈oD₄
ρ₂₆	2	2	-2	-2	0	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	2ζ₈³	2ζ₈	2ζ₈⁵	0	0	0	2ζ₈⁷	0	0	0	0	complex lifted from C₈oD₄
ρ₂₇	2	2	-2	-2	0	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	2ζ₈⁵	2ζ₈⁷	2ζ₈³	0	0	0	2ζ₈	0	0	0	0	complex lifted from C₈oD₄
ρ₂₈	2	2	-2	-2	0	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	2ζ₈⁷	2ζ₈⁵	2ζ₈	0	0	0	2ζ₈³	0	0	0	0	complex lifted from C₈oD₄

Smallest permutation representation of C₈:₉D₄
►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 26 23 11)(2 31 24 16)(3 28 17 13)(4 25 18 10)(5 30 19 15)(6 27 20 12)(7 32 21 9)(8 29 22 14)

(2 6)(4 8)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(18 22)(20 24)

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

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

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

C₈:₉D₄ is a maximal subgroup of
C₄².₂₉₀C₂³ C₄².₂₉₁C₂³ C₄².₂₉₂C₂³ C₄².₂₉₃C₂³ C₄².₂₉₄C₂³ C₂³:₃M₄(2) C₄².₂₉₇C₂³ C₄².₂₉₈C₂³ C₄².₂₉₉C₂³ C₄².₃₀₀C₂³ C₄².₃₀₁C₂³ C₄².₆₉₈C₂³ C₄².₃₀₇C₂³ C₄².₃₀₈C₂³ C₄².₃₀₉C₂³ C₄².₃₁₀C₂³ D₈:₉D₄ SD₁₆:D₄ SD₁₆:₆D₄ D₈:₁₀D₄ SD₁₆:₇D₄ SD₁₆:₈D₄ Q₁₆:₉D₄ Q₁₆:₁₀D₄ C₄².₄₁C₂³ C₄².₄₂C₂³ C₄².₄₃C₂³ C₄².₄₄C₂³ C₄².₄₅C₂³ C₄².₄₆C₂³ C₄².₄₇C₂³ C₄².₄₈C₂³ C₄².₄₉C₂³ C₄².₅₀C₂³ C₄².₅₁C₂³ C₄².₅₂C₂³ C₄².₅₃C₂³ C₄².₅₄C₂³ C₄².₅₅C₂³ C₄².₅₆C₂³ C₄².₅₇C₂³ C₄².₅₈C₂³ C₄².₅₉C₂³ C₄².₆₀C₂³ C₄².₆₁C₂³ C₄².₆₂C₂³ C₄².₆₃C₂³ C₄².₆₄C₂³ C₈:S₄
D_2p:M₄(2): D₄:₆M₄(2) D₄:₇M₄(2) D₄:₈M₄(2) C₈:₉D₁₂ D₆:₂M₄(2) D₆:₃M₄(2) C₂₄:D₄ C₈:₉D₂₀ ...
C_2p.(C₄xD₄): C₄².₂₆₄C₂³ C₄².₂₆₅C₂³ C₄².₂₆₆C₂³ M₄(2):₂₂D₄ D₄xM₄(2) M₄(2):₂₃D₄ C₃:C₈:₂₆D₄ C₄².₄₇D₆ ...
C₈:₉D₄ is a maximal quotient of
C₈:₁₂SD₁₆ C₈:₁₅SD₁₆ C₈:₉Q₁₆ D₄.M₄(2) Q₈.M₄(2) Q₈:₂M₄(2) C₂³.₂₁M₄(2) (C₂xC₈).₁₉₅D₄ C₂³.₂₂M₄(2) C₂³:₂M₄(2) C₄:C₄:₃C₈ (C₂xC₈).Q₈ C₂²:C₄:₄C₈ C₂³.₉M₄(2)
D_2p:M₄(2): C₈:₉D₈ D₄:₂M₄(2) C₈:₉D₁₂ D₆:₂M₄(2) D₆:₃M₄(2) C₂₄:D₄ C₈:₉D₂₀ D₁₀:₄M₄(2) ...
C₄².D_2p: C₄².₂₇Q₈ C₄².₁₀₉D₄ C₄².₄₇D₆ C₄².₄₇D₁₀ C₄².₄₇D₁₄ ...
C_2p.(C₄xD₄): C₂³.₃₆C₄² C₂³.₁₇C₄² C₄:C₈:₁₄C₄ C₃:C₈:₂₆D₄ C₂₄:₃₃D₄ C₅:₂C₈:₂₆D₄ C₄₀:₃₂D₄ C₅:C₈:D₄ ...

Matrix representation of C₈:₉D₄ ►in GL₄(F₁₇) generated by

13	15	0	0
6	4	0	0
0	0	0	1
0	0	4	0

1	0	0	0
13	16	0	0
0	0	0	9
0	0	15	0

1	0	0	0
13	16	0	0
0	0	1	0
0	0	0	16

G:=sub<GL(4,GF(17))| [13,6,0,0,15,4,0,0,0,0,0,4,0,0,1,0],[1,13,0,0,0,16,0,0,0,0,0,15,0,0,9,0],[1,13,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C₈:₉D₄ in GAP, Magma, Sage, TeX

C_8\rtimes_9D_4

% in TeX

G:=Group("C8:9D4");

// GroupNames label

G:=SmallGroup(64,116);

// by ID

G=gap.SmallGroup(64,116);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of C₈:₉D₄ in TeX

G = C8:9D4 order 64 = 26

3rd semidirect product of C8 and D4 acting via D4/C22=C2

G = C₈:₉D₄ order 64 = 2⁶

3^rd semidirect product of C₈ and D₄ acting via D₄/C₂²=C₂