C8.12D4 - GroupNames

G = C₈.₁₂D₄ order 64 = 2⁶

8^th non-split extension by C₈ of D₄ acting via D₄/C₄=C₂

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

Aliases: C₈.₁₂D₄, C₄².₈₂C₂², (C₄×C₈)⋊₉C₂, C₄.₄(C₂×D₄), (C₂×Q₁₆)⋊₅C₂, (C₂×D₈).₃C₂, (C₂×C₄).₅₈D₄, C₄.₄D₄⋊₄C₂, (C₂×SD₁₆)⋊₁₅C₂, C₂.₁₈(C₄○D₈), C₂.₈(C₄⋊₁D₄), (C₂×C₈).₈₀C₂², (C₂×C₄).₁₂₀C₂³, (C₂×D₄).₃₀C₂², C₂².₁₁₆(C₂×D₄), (C₂×Q₈).₂₆C₂², SmallGroup(64,176)

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

Derived series

C₁ — C₂×C₄ — C₈.₁₂D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×C₈ — C₈.₁₂D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₈.₁₂D₄

Upper central

C₁ — C₂² — C₄² — C₈.₁₂D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₈.₁₂D₄

Generators and relations for C₈.₁₂D₄
G = < a,b,c | a⁸=b⁴=1, c²=a⁴, ab=ba, cac^-1=a³, cbc^-1=a⁴b^-1 >

Subgroups: 129 in 65 conjugacy classes, 29 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, C₄.₄D₄, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₈.₁₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄⋊₁D₄, C₄○D₈, C₈.₁₂D₄

Character table of C₈.₁₂D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	8	8	2	2	2	2	2	2	8	8	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	2	0	-2	0	0	0	-2	0	2	0	2	0	-2	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	0	0	0	2	0	-2	0	0	0	2	0	-2	0	-2	0	2	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	2	0	0	0	0	-2	0	2	0	0	0	0	2	0	-2	0	-2	0	2	orthogonal lifted from D₄
ρ₁₃	2	-2	-2	2	0	0	0	0	-2	0	2	0	0	0	0	-2	0	2	0	2	0	-2	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	0	0	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	2	-2	0	0	0	-2i	0	2i	0	0	0	0	-√-2	-√2	√-2	√2	-√-2	-√2	√-2	√2	complex lifted from C₄○D₈
ρ₁₆	2	2	-2	-2	0	0	-2i	0	0	0	0	2i	0	0	√-2	-√2	√-2	-√2	-√-2	√2	-√-2	√2	complex lifted from C₄○D₈
ρ₁₇	2	-2	2	-2	0	0	0	2i	0	-2i	0	0	0	0	√-2	-√2	-√-2	√2	√-2	-√2	-√-2	√2	complex lifted from C₄○D₈
ρ₁₈	2	2	-2	-2	0	0	2i	0	0	0	0	-2i	0	0	√-2	√2	√-2	√2	-√-2	-√2	-√-2	-√2	complex lifted from C₄○D₈
ρ₁₉	2	2	-2	-2	0	0	2i	0	0	0	0	-2i	0	0	-√-2	-√2	-√-2	-√2	√-2	√2	√-2	√2	complex lifted from C₄○D₈
ρ₂₀	2	2	-2	-2	0	0	-2i	0	0	0	0	2i	0	0	-√-2	√2	-√-2	√2	√-2	-√2	√-2	-√2	complex lifted from C₄○D₈
ρ₂₁	2	-2	2	-2	0	0	0	2i	0	-2i	0	0	0	0	-√-2	√2	√-2	-√2	-√-2	√2	√-2	-√2	complex lifted from C₄○D₈
ρ₂₂	2	-2	2	-2	0	0	0	-2i	0	2i	0	0	0	0	√-2	√2	-√-2	-√2	√-2	√2	-√-2	-√2	complex lifted from C₄○D₈

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

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

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

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

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

C₈.₁₂D₄ is a maximal subgroup of
C₄².₄₁₀C₂³ C₄².₄₁₁C₂³ C₄².₅₃₃C₂³
C₈.D_4p: C₈.₃₀D₈ C₈.₃D₈ C₈.₂₁D₈ C₁₆⋊₃D₄ C₈.₇D₈ C₈.₈D₁₂ C₈.₈D₂₀ C₈.₈D₂₈ ...
C₄².D_2p: D₈.₅D₄ Q₁₆.₅D₄ C₈.₂₂SD₁₆ C₈.₁₂SD₁₆ C₄².₃₆₀D₄ M₄(2)⋊₉D₄ C₄².₃₀₈D₄ C₄².₂₆₀D₄ ...
(C_2p×Q₁₆)⋊C₂: C₄².₃₈₇C₂³ Q₁₆⋊₄D₄ Q₁₆⋊₁₂D₄ C₄².₅₂₇C₂³ C₄².₅₃₀C₂³ C₄².₅₃₂C₂³ C₂₄.₂₈D₄ C₄₀.₂₈D₄ ...
(C_2p×SD₁₆)⋊C₂: C₄².₃₈₅C₂³ SD₁₆⋊₂D₄ SD₁₆⋊₁₀D₄ C₄².₅₂₈C₂³ C₄².₅₃₁C₂³ C₂₄.₄₃D₄ C₄₀.₄₃D₄ C₅₆.₄₃D₄ ...
C_4p.(C₂×D₄): M₄(2)⋊₁₀D₄ M₄(2)⋊₁₁D₄ M₄(2).₂₀D₄ D₈⋊₅D₄ D₈⋊₁₂D₄ D₈○SD₁₆ C₂₄.₂₂D₄ C₄₀.₂₂D₄ ...
C₈.₁₂D₄ is a maximal quotient of
C₄².D_2p: C₄².₄₃₃D₄ C₈.₈D₁₂ C₄².₂₁₄D₆ C₈.₈D₂₀ C₄².₂₁₄D₁₀ C₈.₈D₂₈ C₄².₂₁₄D₁₄ ...
(C₂×D₄).D_2p: (C₂²×D₈).C₂ (C₂×C₈).₄₁D₄ C₂₄.₂₂D₄ C₂₄.₄₃D₄ C₄₀.₂₂D₄ C₄₀.₄₃D₄ C₅₆.₂₂D₄ C₅₆.₄₃D₄ ...
(C₂×C₈).D_2p: C₈⋊₅D₈ C₈⋊₅Q₁₆ C₈²⋊₁₂C₂ C₈²⋊₅C₂ C₈.₇Q₁₆ C₈²⋊₃C₂ C₄².₆₆₄C₂³ C₄².₆₆₅C₂³ ...

Matrix representation of C₈.₁₂D₄ ►in GL₄(𝔽₁₇) generated by

12	5	0	0
12	12	0	0
0	0	12	5
0	0	12	12

0	13	0	0
4	0	0	0
0	0	4	0
0	0	0	4

0	13	0	0
13	0	0	0
0	0	4	0
0	0	0	13

G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,12,12,0,0,5,12],[0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4],[0,13,0,0,13,0,0,0,0,0,4,0,0,0,0,13] >;

C₈.₁₂D₄ in GAP, Magma, Sage, TeX

C_8._{12}D_4

% in TeX

G:=Group("C8.12D4");

// GroupNames label

G:=SmallGroup(64,176);

// by ID

G=gap.SmallGroup(64,176);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,230,963,117]);

// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=a^4*b^-1>;

// generators/relations

Export

Character table of C₈.₁₂D₄ in TeX

G = C8.12D4 order 64 = 26

8th non-split extension by C8 of D4 acting via D4/C4=C2

G = C₈.₁₂D₄ order 64 = 2⁶

8^th non-split extension by C₈ of D₄ acting via D₄/C₄=C₂