C8.3D8 - GroupNames

G = C₈.₃D₈ order 128 = 2⁷

3^rd non-split extension by C₈ of D₈ acting via D₈/C₄=C₂²

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₈.₃D₈, D₈.₆D₄, Q₁₆.₆D₄, C₄².₁₄₂D₄, M₅(2).₃C₂², C₈○D₈⋊₃C₂, C₄.₆₇(C₂×D₈), C₈.₇₆(C₂×D₄), C₈.C₈⋊₁C₂, C₈.₁(C₄○D₄), Q₃₂⋊C₂⋊₄C₂, (C₂×C₈).₁₃₀D₄, M₅(2)⋊C₂⋊₃C₂, C₈.₁₇D₄⋊₃C₂, C₁₆⋊C₂².₂C₂, C₈.₁₂D₄⋊₁₂C₂, C₄.₅₅(C₄⋊D₄), C₂.₂₄(C₄⋊D₈), (C₂×C₈).₂₃₅C₂³, (C₄×C₈).₁₆₂C₂², C₄○D₈.₁₈C₂², (C₂×D₈).₄₇C₂², (C₂×Q₁₆).₄₆C₂², C₂².₂₄(C₈⋊C₂²), C₈.C₄.₁₈C₂², (C₂×C₄).₂₈₀(C₂×D₄), SmallGroup(128,944)

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

Derived series

C₁ — C₂×C₈ — C₈.₃D₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₄○D₈ — C₈○D₈ — C₈.₃D₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₈.₃D₈

Upper central

C₁ — C₂ — C₂×C₄ — C₄×C₈ — C₈.₃D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₈.₃D₈

Generators and relations for C₈.₃D₈
G = < a,b,c | a⁸=c²=1, b⁸=a⁴, bab^-1=a^-1, cac=a³, cbc=a⁴b⁷ >

Subgroups: 204 in 77 conjugacy classes, 30 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₆, C₄², C₂²⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, D₈, SD₁₆, Q₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄○D₄, C₄×C₈, C₄≀C₂, C₈.C₄, M₅(2), D₁₆, SD₃₂, Q₃₂, C₄.₄D₄, C₈○D₄, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, M₅(2)⋊C₂, C₈.₁₇D₄, C₈.C₈, C₈○D₈, C₈.₁₂D₄, C₁₆⋊C₂², Q₃₂⋊C₂, C₈.₃D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₈⋊C₂², C₄⋊D₈, C₈.₃D₈

Character table of C₈.₃D₈

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	8A	8B	8C	8D	8E	8F	8G	8H	16A	16B	16C	16D
size	1	1	2	8	16	2	2	4	4	8	16	2	2	2	2	4	4	8	8	8	8	8	8
ρ₁	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	-2	2	0	-2	2	0	0	-2	0	2	0	0	2	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	-2	0	-2	2	0	0	2	0	2	0	0	2	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	0	0	2	2	-2	-2	0	0	-2	2	2	-2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	0	0	2	2	2	2	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	0	0	2	-2	0	0	0	0	0	-2	-2	0	2	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	-2	0	0	2	-2	0	0	0	0	0	2	2	0	-2	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	2	-2	0	0	2	-2	0	0	0	0	0	2	2	0	-2	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₆	2	2	-2	0	0	2	-2	0	0	0	0	0	-2	-2	0	2	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	0	0	-2	2	0	0	0	0	-2	0	0	-2	0	2	2i	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	2	-2	0	0	-2	2	0	0	0	0	-2	0	0	-2	0	2	-2i	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	4	4	4	0	0	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₀	4	-4	0	0	0	0	0	2i	-2i	0	0	-2√2	2√-2	-2√-2	2√2	0	0	0	0	0	0	0	0	complex faithful
ρ₂₁	4	-4	0	0	0	0	0	2i	-2i	0	0	2√2	-2√-2	2√-2	-2√2	0	0	0	0	0	0	0	0	complex faithful
ρ₂₂	4	-4	0	0	0	0	0	-2i	2i	0	0	-2√2	-2√-2	2√-2	2√2	0	0	0	0	0	0	0	0	complex faithful
ρ₂₃	4	-4	0	0	0	0	0	-2i	2i	0	0	2√2	2√-2	-2√-2	-2√2	0	0	0	0	0	0	0	0	complex faithful

Smallest permutation representation of C₈.₃D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

Matrix representation of C₈.₃D₈ ►in GL₄(𝔽₁₇) generated by

10	7	0	0
5	0	0	0
0	0	7	10
0	0	12	0

0	0	0	6
0	0	3	0
1	0	0	0
1	16	0	0

1	0	0	0
1	16	0	0
0	0	0	6
0	0	3	0

G:=sub<GL(4,GF(17))| [10,5,0,0,7,0,0,0,0,0,7,12,0,0,10,0],[0,0,1,1,0,0,0,16,0,3,0,0,6,0,0,0],[1,1,0,0,0,16,0,0,0,0,0,3,0,0,6,0] >;

C₈.₃D₈ in GAP, Magma, Sage, TeX

C_8._3D_8

% in TeX

G:=Group("C8.3D8");

// GroupNames label

G:=SmallGroup(128,944);

// by ID

G=gap.SmallGroup(128,944);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,2019,248,1684,438,242,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=a^4*b^7>;

// generators/relations

Export

Character table of C₈.₃D₈ in TeX

G = C8.3D8 order 128 = 27

3rd non-split extension by C8 of D8 acting via D8/C4=C22

G = C₈.₃D₈ order 128 = 2⁷

3^rd non-split extension by C₈ of D₈ acting via D₈/C₄=C₂²