C8:4D4 - GroupNames

G = C₈⋊₄D₄ order 64 = 2⁶

1^st semidirect product of C₈ and D₄ acting via D₄/C₄=C₂

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

Aliases: C₈⋊₄D₄, C₄⋊₁D₈, C₄².₈₀C₂², (C₄×C₈)⋊₈C₂, (C₂×D₈)⋊₅C₂, C₄.₂(C₂×D₄), C₄⋊₁D₄⋊₃C₂, C₂.₁₀(C₂×D₈), (C₂×C₄).₇₇D₄, C₂.₆(C₄⋊₁D₄), (C₂×C₈).₇₈C₂², (C₂×C₄).₁₁₈C₂³, (C₂×D₄).₂₉C₂², C₂².₁₁₄(C₂×D₄), SmallGroup(64,174)

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

Derived series

C₁ — C₂×C₄ — C₈⋊₄D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×C₈ — C₈⋊₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₈⋊₄D₄

Upper central

C₁ — C₂² — C₄² — C₈⋊₄D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₈⋊₄D₄

Generators and relations for C₈⋊₄D₄
G = < a,b,c | a⁸=b⁴=c²=1, ab=ba, cac=a^-1, cbc=b^-1 >

Subgroups: 193 in 81 conjugacy classes, 33 normal (7 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₄², C₂×C₈, D₈, C₂×D₄, C₂×D₄, C₄×C₈, C₄⋊₁D₄, C₂×D₈, C₈⋊₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄⋊₁D₄, C₂×D₈, C₈⋊₄D₄

Character table of C₈⋊₄D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	8	8	8	8	2	2	2	2	2	2	2	2	2	2	2	2	2	2
ρ₁	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	-2	-2	2	0	0	0	0	0	0	-2	0	2	0	0	2	0	2	0	-2	0	-2	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	0	0	0	0	0	0	-2	0	2	0	0	-2	0	-2	0	2	0	2	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	0	0	0	0	0	2	0	-2	0	2	0	2	0	-2	0	-2	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	2	0	0	0	0	0	0	2	0	-2	0	-2	0	-2	0	2	0	2	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	0	0	0	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	0	0	0	0	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	2	-2	0	0	0	0	0	-2	0	2	0	0	-√2	√2	√2	-√2	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₆	2	2	-2	-2	0	0	0	0	2	0	0	0	0	-2	-√2	√2	√2	-√2	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	0	0	0	0	2	0	0	0	0	-2	√2	-√2	-√2	√2	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₈	2	-2	2	-2	0	0	0	0	0	-2	0	2	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₉	2	-2	2	-2	0	0	0	0	0	2	0	-2	0	0	√2	√2	-√2	-√2	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₂₀	2	2	-2	-2	0	0	0	0	-2	0	0	0	0	2	√2	√2	-√2	-√2	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₂₁	2	-2	2	-2	0	0	0	0	0	2	0	-2	0	0	-√2	-√2	√2	√2	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₂₂	2	2	-2	-2	0	0	0	0	-2	0	0	0	0	2	-√2	-√2	√2	√2	√2	√2	-√2	-√2	orthogonal lifted from D₈

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

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

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

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

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

C₈⋊₄D₄ is a maximal subgroup of
C₈.₂₄D₈ C₄.D₁₆ C₄⋊₁D₄⋊C₄ D₄⋊D₈ C₄².₁₈₁C₂³ Q₈⋊D₈ C₈⋊₁₃SD₁₆ C₈⋊₂SD₁₆ C₈²⋊₅C₂ D₈⋊₂D₄ Q₁₆⋊₂D₄ D₈⋊₃D₄ C₄.₄D₁₆ C₈.₁₃SD₁₆ C₄².₂₆₃D₄ D₈○D₈ Q₈⋊₅D₈ C₄².₅₃₀C₂³
C_8p⋊D₄: C₄⋊D₁₆ C₁₆⋊₅D₄ C₁₆⋊₃D₄ C₁₂⋊₄D₈ C₂₄⋊₅D₄ C₂₀⋊₄D₈ C₄₀⋊₅D₄ C₂₈⋊₄D₈ ...
C_4p⋊D₈: C₈⋊₇D₈ C₈⋊₂D₈ C₈⋊₅D₈ C₈⋊₄D₈ C₈⋊₃D₈ C₁₂⋊D₈ C₂₀⋊D₈ C₂₈⋊D₈ ...
C₈⋊_pD₄⋊C₂: C₈⋊₆SD₁₆ C₄².₆₆₄C₂³ C₄².₃₆₀D₄ M₄(2)⋊₇D₄ M₄(2)⋊₁₁D₄ C₄².₃₆₆D₄ C₄².₃₈₈C₂³ C₄².₂₆₁D₄ ...
C₈⋊₄D₄ is a maximal quotient of
C₈²⋊₅C₂ C₈⋊₄Q₁₆ C₈.₂D₈ C₄².₅₉Q₈ C₄².₄₃₂D₄ (C₂×C₄)⋊₆D₈ (C₂×C₄)⋊₂D₈ (C₂×C₄).₂₇D₈ C₄⋊Q₃₂ C₈.₂₁D₈ C₈.₇D₈
C_8p⋊D₄: C₄⋊D₁₆ C₁₆⋊₅D₄ C₁₆⋊₃D₄ C₁₂⋊₄D₈ C₂₄⋊₅D₄ C₂₀⋊₄D₈ C₄₀⋊₅D₄ C₂₈⋊₄D₈ ...
C_4p⋊D₈: C₈⋊₅D₈ C₈⋊₄D₈ C₈⋊₃D₈ C₁₂⋊D₈ C₂₀⋊D₈ C₂₈⋊D₈ ...

Matrix representation of C₈⋊₄D₄ ►in GL₄(𝔽₁₇) generated by

3	14	0	0
3	3	0	0
0	0	3	14
0	0	3	3

16	0	0	0
0	16	0	0
0	0	0	1
0	0	16	0

1	0	0	0
0	16	0	0
0	0	16	0
0	0	0	1

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

C₈⋊₄D₄ in GAP, Magma, Sage, TeX

C_8\rtimes_4D_4

% in TeX

G:=Group("C8:4D4");

// GroupNames label

G:=SmallGroup(64,174);

// by ID

G=gap.SmallGroup(64,174);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,86,963,117]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈⋊₄D₄ in TeX

G = C8⋊4D4 order 64 = 26

1st semidirect product of C8 and D4 acting via D4/C4=C2

G = C₈⋊₄D₄ order 64 = 2⁶

1^st semidirect product of C₈ and D₄ acting via D₄/C₄=C₂