C4:D8 - GroupNames

G = C₄:D₈ order 64 = 2⁶

The semidirect product of C₄ and D₈ acting via D₈/D₄=C₂

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄:₂D₈, D₄:₁D₄, C₄².₁₆C₂², C₄:C₈:₃C₂, (C₄xD₄):₃C₂, (C₂xD₈):₃C₂, C₂.₅(C₂xD₈), C₄:₁D₄:₂C₂, (C₂xC₄).₂₆D₄, C₄.₂₉(C₂xD₄), D₄:C₄:₆C₂, (C₂xC₈).₃C₂², C₄.₃₉(C₄oD₄), C₄:C₄.₅₆C₂², C₂.₉(C₈:C₂²), (C₂xC₄).₈₇C₂³, C₂².₈₃(C₂xD₄), C₂.₁₁(C₄:D₄), (C₂xD₄).₁₁C₂², SmallGroup(64,140)

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

Derived series

C₁ — C₂xC₄ — C₄:D₈

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂xD₄ — C₄xD₄ — C₄:D₈

Lower central

C₁ — C₂ — C₂xC₄ — C₄:D₈

Upper central

C₁ — C₂² — C₄² — C₄:D₈

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₄:D₈

Generators and relations for C₄:D₈
G = < a,b,c | a⁴=b⁸=c²=1, bab^-1=cac=a^-1, cbc=b^-1 >

Subgroups: 157 in 70 conjugacy classes, 29 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, D₄, D₄, C₂³, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, D₈, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, D₄:C₄, C₄:C₈, C₄xD₄, C₄:₁D₄, C₂xD₈, C₄:D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂xD₄, C₄oD₄, C₄:D₄, C₂xD₈, C₈:C₂², C₄:D₈

Character table of C₄:D₈

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D
size	1	1	1	1	4	4	8	8	2	2	2	2	4	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	trivial
ρ₂	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	linear of order 2
ρ₄	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	linear of order 2
ρ₆	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	linear of order 2
ρ₈	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	0	0	0	0	-2	-2	2	2	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	0	-2	-2	-2	-2	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	-2	2	0	0	2	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	2	-2	2	-2	0	0	2	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	-2	-2	2	0	0	0	0	0	0	2	-2	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₄	2	-2	-2	2	0	0	0	0	0	0	2	-2	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₅	2	-2	-2	2	0	0	0	0	0	0	-2	2	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₆	2	-2	-2	2	0	0	0	0	0	0	-2	2	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₇	2	-2	2	-2	0	0	0	0	-2	2	0	0	2i	-2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₁₈	2	-2	2	-2	0	0	0	0	-2	2	0	0	-2i	2i	0	0	0	0	0	complex lifted from C₄oD₄
ρ₁₉	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈:C₂²

Smallest permutation representation of C₄:D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

C₄:D₈ is a maximal subgroup of
D₄:D₈ Q₈:D₈ D₄:₃D₈ Q₈:₃D₈ C₄².₁₈₉C₂³ D₄.₂SD₁₆ D₄.₂D₈ C₄².₂₄₈C₂³ C₄².₂₅₂C₂³ C₄².₄₄₃D₄ C₄².₄₄₄D₄ C₄².₁₈C₂³ C₄².₂₂₁D₄ C₄².₄₅₀D₄ C₄².₂₂₇D₄ C₄².₂₃₃D₄ C₄².₃₅₃C₂³ C₄².₃₅₆C₂³ C₄².₂₆₃D₄ C₄².₂₇₀D₄ Q₈:₄D₈ C₄².₅₀₂C₂³ C₄².₅₀₇C₂³ C₄².₅₁₁C₂³ C₄².₅₁₄C₂³ Q₈:₅D₈
D_4p:D₄: D₈:₉D₄ D₈:₄D₄ D₁₂:₃D₄ C₄:D₂₄ C₁₂:₂D₈ D₂₀:₃D₄ C₄:D₄₀ C₂₀:₂D₈ ...
C_4p:D₈: C₈:₈D₈ C₈:₇D₈ C₈:D₈ C₈:₂D₈ C₁₂:₇D₈ C₂₀:₇D₈ C₂₈:₇D₈ ...
(C_pxD₄):D₄: C₄².₂₁₁D₄ C₄².₁₄C₂³ Dic₃:D₈ Dic₅:D₈ Dic₇:D₈ ...
C₈:_pD₄:C₂: D₄:SD₁₆ C₄².₁₈₅C₂³ C₄².₂₇₂D₄ C₄².₂₇₅D₄ C₄².₄₀₆C₂³ C₄².₄₁₀C₂³ C₄².₂₉₃D₄ C₄².₂₉₅D₄ ...
C₄:D₈ is a maximal quotient of
C₈.₂₈D₈ C₈.D₈ C₄².₉₈D₄ (C₂xC₄):₉D₈ D₄:C₄:C₄ C₄².₂₉Q₈ C₄².₁₁₈D₄ C₄:C₄:₇D₄ (C₂xC₄):₃D₈ (C₂xC₄).₂₃D₈ (C₂xC₄).₂₄D₈ (C₂xC₈).₁Q₈ Q₁₆:₂D₄ D₈.₄D₄ Q₁₆.₄D₄ D₈.₅D₄ Q₁₆.₅D₄ C₈.₃D₈ C₈.₅D₈
D_4p:D₄: D₈:₂D₄ D₈:₃D₄ D₁₂:₃D₄ C₄:D₂₄ C₁₂:₂D₈ D₂₀:₃D₄ C₄:D₄₀ C₂₀:₂D₈ ...
C_4p:D₈: C₈:₈D₈ C₈:₇D₈ C₈:D₈ C₈:₂D₈ C₁₂:₇D₈ C₂₀:₇D₈ C₂₈:₇D₈ ...
(C_pxD₄):D₄: (C₂xC₄):₂D₈ Dic₃:D₈ Dic₅:D₈ Dic₇:D₈ ...

Matrix representation of C₄:D₈ ►in GL₄(F₁₇) generated by

13	9	0	0
0	4	0	0
0	0	16	0
0	0	0	16

13	0	0	0
4	4	0	0
0	0	3	14
0	0	3	3

13	9	0	0
4	4	0	0
0	0	16	0
0	0	0	1

G:=sub<GL(4,GF(17))| [13,0,0,0,9,4,0,0,0,0,16,0,0,0,0,16],[13,4,0,0,0,4,0,0,0,0,3,3,0,0,14,3],[13,4,0,0,9,4,0,0,0,0,16,0,0,0,0,1] >;

C₄:D₈ in GAP, Magma, Sage, TeX

C_4\rtimes D_8

% in TeX

G:=Group("C4:D8");

// GroupNames label

G:=SmallGroup(64,140);

// by ID

G=gap.SmallGroup(64,140);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,1444,376,88]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄:D₈ in TeX

G = C4:D8 order 64 = 26

The semidirect product of C4 and D8 acting via D8/D4=C2

G = C₄:D₈ order 64 = 2⁶

The semidirect product of C₄ and D₈ acting via D₈/D₄=C₂