Q16.8D4 - GroupNames

G = Q₁₆.₈D₄ order 128 = 2⁷

1^st non-split extension by Q₁₆ of D₄ acting via D₄/C₂²=C₂

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

Aliases: Q₁₆.₈D₄, C₂²⋊₂Q₃₂, C₂³.₄₇D₈, (C₂×Q₃₂)⋊₁C₂, (C₂×C₄).₃₃D₈, (C₂×C₈).₆₄D₄, C₈.₆₄(C₂×D₄), C₂.₄(C₂×Q₃₂), C₂.Q₃₂⋊₅C₂, C₄.₂₀C₂²≀C₂, C₂²⋊C₁₆.₅C₂, (C₂×C₁₆).₂C₂², C₂².₉₈(C₂×D₈), C₄.₁₂(C₈⋊C₂²), (C₂×C₈).₅₁₂C₂³, C₂.D₈.₃C₂², C₈.₁₈D₄.₃C₂, C₂.₇(Q₃₂⋊C₂), (C₂²×C₄).₃₄₇D₄, (C₂×Q₁₆).₁C₂², (C₂²×Q₁₆).₇C₂, C₂.₂₈(C₂²⋊D₈), (C₂²×C₈).₁₂₈C₂², (C₂×C₄).₇₈₀(C₂×D₄), SmallGroup(128,920)

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

Derived series

C₁ — C₂×C₈ — Q₁₆.₈D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂²×Q₁₆ — Q₁₆.₈D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — Q₁₆.₈D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — Q₁₆.₈D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — Q₁₆.₈D₄

Generators and relations for Q₁₆.₈D₄
G = < a,b,c,d | a⁸=1, b²=d²=a⁴, c⁴=a², bab^-1=dad^-1=a^-1, ac=ca, cbc^-1=ab, dbd^-1=a^-1b, dcd^-1=a⁶c³ >

Subgroups: 244 in 108 conjugacy classes, 36 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, Q₈⋊C₄, C₂.D₈, C₂×C₁₆, Q₃₂, C₂²⋊Q₈, C₂²×C₈, C₂×Q₁₆, C₂×Q₁₆, C₂×Q₁₆, C₂²×Q₈, C₂²⋊C₁₆, C₂.Q₃₂, C₈.₁₈D₄, C₂×Q₃₂, C₂²×Q₁₆, Q₁₆.₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, Q₃₂, C₂²≀C₂, C₂×D₈, C₈⋊C₂², C₂²⋊D₈, C₂×Q₃₂, Q₃₂⋊C₂, Q₁₆.₈D₄

Character table of Q₁₆.₈D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	2	2	4	8	8	8	8	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	-2	-2	2	2	-2	0	0	0	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	0	0	2	-2	0	-2	0	0	2	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	2	0	0	-2	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	2	2	2	0	0	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	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
ρ₁₅	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	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	-2	-2	-2	0	0	0	0	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	-2	-2	-2	0	0	0	0	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	-2	-2	2	0	0	0	0	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	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	symplectic lifted from Q₃₂, Schur index 2
ρ₂₀	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₁	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₂	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	symplectic lifted from Q₃₂, Schur index 2
ρ₂₃	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	-√2	√2	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₄	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	√2	-√2	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	symplectic lifted from Q₃₂, Schur index 2
ρ₂₅	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	√2	-√2	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₂₆	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	-√2	√2	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	symplectic lifted from Q₃₂, Schur index 2
ρ₂₇	4	-4	-4	4	0	0	-4	4	0	0	0	0	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	symplectic lifted from Q₃₂⋊C₂, Schur index 2
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	symplectic lifted from Q₃₂⋊C₂, Schur index 2

Smallest permutation representation of Q₁₆.₈D₄
►On 64 points

Generators in S₆₄

(1 21 5 25 9 29 13 17)(2 22 6 26 10 30 14 18)(3 23 7 27 11 31 15 19)(4 24 8 28 12 32 16 20)(33 58 37 62 41 50 45 54)(34 59 38 63 42 51 46 55)(35 60 39 64 43 52 47 56)(36 61 40 49 44 53 48 57)

(1 35 9 43)(2 57 10 49)(3 33 11 41)(4 55 12 63)(5 47 13 39)(6 53 14 61)(7 45 15 37)(8 51 16 59)(17 60 25 52)(18 36 26 44)(19 58 27 50)(20 34 28 42)(21 56 29 64)(22 48 30 40)(23 54 31 62)(24 46 32 38)

(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)

(1 51 9 59)(2 50 10 58)(3 49 11 57)(4 64 12 56)(5 63 13 55)(6 62 14 54)(7 61 15 53)(8 60 16 52)(17 46 25 38)(18 45 26 37)(19 44 27 36)(20 43 28 35)(21 42 29 34)(22 41 30 33)(23 40 31 48)(24 39 32 47)

G:=sub<Sym(64)| (1,21,5,25,9,29,13,17)(2,22,6,26,10,30,14,18)(3,23,7,27,11,31,15,19)(4,24,8,28,12,32,16,20)(33,58,37,62,41,50,45,54)(34,59,38,63,42,51,46,55)(35,60,39,64,43,52,47,56)(36,61,40,49,44,53,48,57), (1,35,9,43)(2,57,10,49)(3,33,11,41)(4,55,12,63)(5,47,13,39)(6,53,14,61)(7,45,15,37)(8,51,16,59)(17,60,25,52)(18,36,26,44)(19,58,27,50)(20,34,28,42)(21,56,29,64)(22,48,30,40)(23,54,31,62)(24,46,32,38), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,51,9,59)(2,50,10,58)(3,49,11,57)(4,64,12,56)(5,63,13,55)(6,62,14,54)(7,61,15,53)(8,60,16,52)(17,46,25,38)(18,45,26,37)(19,44,27,36)(20,43,28,35)(21,42,29,34)(22,41,30,33)(23,40,31,48)(24,39,32,47)>;

G:=Group( (1,21,5,25,9,29,13,17)(2,22,6,26,10,30,14,18)(3,23,7,27,11,31,15,19)(4,24,8,28,12,32,16,20)(33,58,37,62,41,50,45,54)(34,59,38,63,42,51,46,55)(35,60,39,64,43,52,47,56)(36,61,40,49,44,53,48,57), (1,35,9,43)(2,57,10,49)(3,33,11,41)(4,55,12,63)(5,47,13,39)(6,53,14,61)(7,45,15,37)(8,51,16,59)(17,60,25,52)(18,36,26,44)(19,58,27,50)(20,34,28,42)(21,56,29,64)(22,48,30,40)(23,54,31,62)(24,46,32,38), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,51,9,59)(2,50,10,58)(3,49,11,57)(4,64,12,56)(5,63,13,55)(6,62,14,54)(7,61,15,53)(8,60,16,52)(17,46,25,38)(18,45,26,37)(19,44,27,36)(20,43,28,35)(21,42,29,34)(22,41,30,33)(23,40,31,48)(24,39,32,47) );

G=PermutationGroup([[(1,21,5,25,9,29,13,17),(2,22,6,26,10,30,14,18),(3,23,7,27,11,31,15,19),(4,24,8,28,12,32,16,20),(33,58,37,62,41,50,45,54),(34,59,38,63,42,51,46,55),(35,60,39,64,43,52,47,56),(36,61,40,49,44,53,48,57)], [(1,35,9,43),(2,57,10,49),(3,33,11,41),(4,55,12,63),(5,47,13,39),(6,53,14,61),(7,45,15,37),(8,51,16,59),(17,60,25,52),(18,36,26,44),(19,58,27,50),(20,34,28,42),(21,56,29,64),(22,48,30,40),(23,54,31,62),(24,46,32,38)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,51,9,59),(2,50,10,58),(3,49,11,57),(4,64,12,56),(5,63,13,55),(6,62,14,54),(7,61,15,53),(8,60,16,52),(17,46,25,38),(18,45,26,37),(19,44,27,36),(20,43,28,35),(21,42,29,34),(22,41,30,33),(23,40,31,48),(24,39,32,47)]])

Matrix representation of Q₁₆.₈D₄ ►in GL₄(𝔽₁₇) generated by

16	0	0	0
0	16	0	0
0	0	14	3
0	0	14	14

1	0	0	0
0	16	0	0
0	0	12	5
0	0	5	5

0	16	0	0
16	0	0	0
0	0	13	6
0	0	11	13

0	1	0	0
1	0	0	0
0	0	1	7
0	0	7	16

G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,14,0,0,3,14],[1,0,0,0,0,16,0,0,0,0,12,5,0,0,5,5],[0,16,0,0,16,0,0,0,0,0,13,11,0,0,6,13],[0,1,0,0,1,0,0,0,0,0,1,7,0,0,7,16] >;

Q₁₆.₈D₄ in GAP, Magma, Sage, TeX

Q_{16}._8D_4

% in TeX

G:=Group("Q16.8D4");

// GroupNames label

G:=SmallGroup(128,920);

// by ID

G=gap.SmallGroup(128,920);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,456,422,1123,570,360,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d|a^8=1,b^2=d^2=a^4,c^4=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^6*c^3>;

// generators/relations

Export

Character table of Q₁₆.₈D₄ in TeX

G = Q16.8D4 order 128 = 27

1st non-split extension by Q16 of D4 acting via D4/C22=C2

G = Q₁₆.₈D₄ order 128 = 2⁷

1^st non-split extension by Q₁₆ of D₄ acting via D₄/C₂²=C₂