D8.9D4 - GroupNames

G = D₈.₉D₄ order 128 = 2⁷

1^st non-split extension by D₈ of D₄ acting via D₄/C₂²=C₂

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

Aliases: D₈.₉D₄, C₂²⋊₂SD₃₂, C₂³.₄₆D₈, (C₂×C₈).₆₃D₄, C₈.₆₃(C₂×D₄), (C₂×C₄).₃₂D₈, (C₂×C₁₆)⋊₇C₂², (C₂×SD₃₂)⋊₉C₂, C₂.D₁₆⋊₁₀C₂, C₂.₆(C₂×SD₃₂), C₂²⋊C₁₆⋊₁₁C₂, C₂.D₈⋊₂C₂², C₄.₁₉C₂²≀C₂, C₈.₁₈D₄⋊₁C₂, (C₂×Q₁₆)⋊₁C₂², (C₂²×D₈).₇C₂, C₂².₉₇(C₂×D₈), C₄.₁₁(C₈⋊C₂²), C₂.₈(C₁₆⋊C₂²), (C₂×C₈).₅₁₁C₂³, (C₂²×C₄).₃₄₆D₄, C₂.₂₇(C₂²⋊D₈), (C₂×D₈).₁₀₈C₂², (C₂²×C₈).₁₂₇C₂², (C₂×C₄).₇₇₉(C₂×D₄), 2-Sylow(CO-(4,7)), SmallGroup(128,919)

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

Derived series

C₁ — C₂×C₈ — D₈.₉D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂²×D₈ — D₈.₉D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — D₈.₉D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — D₈.₉D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — D₈.₉D₄

Generators and relations for D₈.₉D₄
G = < a,b,c,d | a⁸=b²=1, c⁴=a², d²=a⁴, bab=dad^-1=a^-1, ac=ca, cbc^-1=ab, dbd^-1=a⁵b, dcd^-1=a²c³ >

Subgroups: 404 in 128 conjugacy classes, 36 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₁₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, D₈, Q₁₆, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, Q₈⋊C₄, C₂.D₈, C₂×C₁₆, SD₃₂, C₂²⋊Q₈, C₂²×C₈, C₂×D₈, C₂×D₈, C₂×Q₁₆, C₂²×D₄, C₂²⋊C₁₆, C₂.D₁₆, C₈.₁₈D₄, C₂×SD₃₂, C₂²×D₈, D₈.₉D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, SD₃₂, C₂²≀C₂, C₂×D₈, C₈⋊C₂², C₂²⋊D₈, C₂×SD₃₂, C₁₆⋊C₂², D₈.₉D₄

Character table of D₈.₉D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	8	8	8	8	2	2	4	16	16	2	2	2	2	4	4	4	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	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	-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	-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	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	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	-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	-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	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	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	0	0	0	0	2	-2	2	-2	0	0	0	-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	-2	2	0	0	2	-2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	2	0	0	2	-2	0	0	2	-2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	0	0	0	0	2	2	-2	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	2	2	0	0	0	0	2	2	2	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	2	2	0	0	0	0	-2	-2	-2	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₆	2	2	2	2	-2	-2	0	0	0	0	-2	-2	2	0	0	0	0	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₇	2	2	2	2	-2	-2	0	0	0	0	-2	-2	2	0	0	0	0	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₈	2	2	2	2	2	2	0	0	0	0	-2	-2	-2	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₉	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₀	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₁	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₂	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₃	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₄	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₅	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₆	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₇	4	-4	-4	4	0	0	0	0	0	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
ρ₂₈	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	2√2	-2√2	2√2	-2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²
ρ₂₉	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2√2	2√2	-2√2	2√2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₁₆⋊C₂²

Smallest permutation representation of D₈.₉D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

Matrix representation of D₈.₉D₄ ►in GL₄(𝔽₁₇) generated by

0	6	0	0
14	6	0	0
0	0	16	0
0	0	0	16

16	2	0	0
0	1	0	0
0	0	16	0
0	0	0	1

11	15	0	0
1	9	0	0
0	0	0	16
0	0	16	0

11	15	0	0
10	6	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(17))| [0,14,0,0,6,6,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,2,1,0,0,0,0,16,0,0,0,0,1],[11,1,0,0,15,9,0,0,0,0,0,16,0,0,16,0],[11,10,0,0,15,6,0,0,0,0,0,1,0,0,1,0] >;

D₈.₉D₄ in GAP, Magma, Sage, TeX

D_8._9D_4

% in TeX

G:=Group("D8.9D4");

// GroupNames label

G:=SmallGroup(128,919);

// by ID

G=gap.SmallGroup(128,919);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,1123,570,360,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^2,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a*b,d*b*d^-1=a^5*b,d*c*d^-1=a^2*c^3>;

// generators/relations

Export

Character table of D₈.₉D₄ in TeX

G = D8.9D4 order 128 = 27

1st non-split extension by D8 of D4 acting via D4/C22=C2

G = D₈.₉D₄ order 128 = 2⁷

1^st non-split extension by D₈ of D₄ acting via D₄/C₂²=C₂