C8.5D8 - GroupNames

G = C₈.₅D₈ order 128 = 2⁷

5^th non-split extension by C₈ of D₈ acting via D₈/C₄=C₂²

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

Aliases: C₈.₅D₈, D₈.₇D₄, Q₁₆.₇D₄, C₄².₁₄₄D₄, M₅(2).₅C₂², C₈○D₈.₄C₂, C₄.₆₉(C₂×D₈), C₈.₇₈(C₂×D₄), C₈.C₈⋊₃C₂, C₈.₃(C₄○D₄), (C₂×C₈).₁₃₂D₄, C₄⋊Q₁₆⋊₁₆C₂, C₈.₁₇D₄⋊₄C₂, Q₃₂⋊C₂.₂C₂, C₄.₅₇(C₄⋊D₄), C₂.₂₆(C₄⋊D₈), (C₂×C₈).₂₃₇C₂³, (C₄×C₈).₁₆₄C₂², C₄○D₈.₂₀C₂², (C₂×Q₁₆).₄₇C₂², C₂².₂₆(C₈⋊C₂²), C₈.C₄.₂₀C₂², (C₂×C₄).₂₈₂(C₂×D₄), SmallGroup(128,946)

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

Derived series

C₁ — C₂×C₈ — C₈.₅D₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₄○D₈ — C₈○D₈ — C₈.₅D₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₈.₅D₈

Upper central

C₁ — C₂ — C₂×C₄ — C₄×C₈ — C₈.₅D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₈.₅D₈

Generators and relations for C₈.₅D₈
G = < a,b,c | a⁸=1, b⁸=c²=a⁴, bab^-1=a³, cac^-1=a^-1, cbc^-1=b⁷ >

Subgroups: 172 in 75 conjugacy classes, 30 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₁₆, C₄², C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, Q₁₆, C₂×Q₈, C₄○D₄, C₄×C₈, C₄≀C₂, C₈.C₄, M₅(2), SD₃₂, Q₃₂, C₄⋊Q₈, C₈○D₄, C₂×Q₁₆, C₂×Q₁₆, C₄○D₈, C₈.₁₇D₄, C₈.C₈, C₈○D₈, C₄⋊Q₁₆, Q₃₂⋊C₂, C₈.₅D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₈⋊C₂², C₄⋊D₈, C₈.₅D₈

Character table of C₈.₅D₈

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D	8E	8F	8G	8H	16A	16B	16C	16D
size	1	1	2	8	2	2	4	4	8	16	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	0	2	2	-2	-2	0	0	0	-2	-2	2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	2	-2	2	0	0	-2	0	0	2	2	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	-2	-2	2	0	0	2	0	0	2	2	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	0	2	2	2	2	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	0	2	-2	0	0	0	0	0	0	0	2	2	-2	0	0	0	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	-2	0	2	-2	0	0	0	0	0	0	0	2	2	-2	0	0	0	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	2	-2	0	2	-2	0	0	0	0	0	0	0	-2	-2	2	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₆	2	2	-2	0	2	-2	0	0	0	0	0	0	0	-2	-2	2	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₇	2	2	-2	0	-2	2	0	0	0	0	0	-2	-2	0	0	0	2	-2i	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	2	-2	0	-2	2	0	0	0	0	0	-2	-2	0	0	0	2	2i	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	4	4	4	0	-4	-4	0	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	-2	2	0	0	0	-2√2	2√2	2√2	-2√2	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₂₁	4	-4	0	0	0	0	2	-2	0	0	0	2√2	-2√2	2√2	-2√2	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₂₂	4	-4	0	0	0	0	-2	2	0	0	0	2√2	-2√2	-2√2	2√2	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₂₃	4	-4	0	0	0	0	2	-2	0	0	0	-2√2	2√2	-2√2	2√2	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of C₈.₅D₈
►On 32 points

Generators in S₃₂

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

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

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

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

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

Matrix representation of C₈.₅D₈ ►in GL₄(𝔽₇) generated by

5	0	0	2
1	3	3	4
1	0	0	1
5	3	6	6

3	0	6	6
4	5	2	3
1	6	1	5
3	1	5	5

3	0	6	1
1	1	0	2
1	6	1	5
5	6	4	2

G:=sub<GL(4,GF(7))| [5,1,1,5,0,3,0,3,0,3,0,6,2,4,1,6],[3,4,1,3,0,5,6,1,6,2,1,5,6,3,5,5],[3,1,1,5,0,1,6,6,6,0,1,4,1,2,5,2] >;

C₈.₅D₈ in GAP, Magma, Sage, TeX

C_8._5D_8

% in TeX

G:=Group("C8.5D8");

// GroupNames label

G:=SmallGroup(128,946);

// by ID

G=gap.SmallGroup(128,946);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,512,422,2019,248,1684,438,242,4037,1027,124]);

// Polycyclic

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

// generators/relations

Export

Character table of C₈.₅D₈ in TeX

G = C8.5D8 order 128 = 27

5th non-split extension by C8 of D8 acting via D8/C4=C22

G = C₈.₅D₈ order 128 = 2⁷

5^th non-split extension by C₈ of D₈ acting via D₈/C₄=C₂²