D8:Q8 - GroupNames

G = D₈⋊Q₈ order 128 = 2⁷

2^nd semidirect product of D₈ and Q₈ acting via Q₈/C₄=C₂

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

Aliases: D₈⋊₂Q₈, C₄.₁₀SD₃₂, C₄².₁₄₇D₄, C₄⋊C₁₆⋊₁₂C₂, (C₄×D₈).₇C₂, C₁₆⋊₄C₄⋊₇C₂, (C₂×C₈).₇₁D₄, C₈.₃₁(C₂×Q₈), C₈⋊₂Q₈⋊₁₀C₂, (C₂×C₄).₁₅₃D₈, C₂.D₁₆.₅C₂, C₈.₆₅(C₄○D₄), (C₄×C₈).₆₇C₂², C₂.₁₁(C₂×SD₃₂), (C₂×C₁₆).₄₃C₂², (C₂×C₈).₅₂₈C₂³, C₂².₁₁₄(C₂×D₈), C₄.₄₇(C₂²⋊Q₈), C₂.₁₄(C₁₆⋊C₂²), C₂.D₈.₁₃C₂², C₂.₁₄(D₄⋊Q₈), (C₂×D₈).₁₁₃C₂², C₄.₁₀(C₈.C₂²), (C₂×C₄).₇₉₆(C₂×D₄), SmallGroup(128,958)

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

Derived series

C₁ — C₂×C₈ — D₈⋊Q₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×D₈ — C₄×D₈ — D₈⋊Q₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — D₈⋊Q₈

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — D₈⋊Q₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — D₈⋊Q₈

Generators and relations for D₈⋊Q₈
G = < a,b,c,d | a⁸=b²=c⁴=1, d²=c², bab=dad^-1=a^-1, ac=ca, cbc^-1=a⁴b, dbd^-1=a⁵b, dcd^-1=c^-1 >

Subgroups: 196 in 71 conjugacy classes, 34 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, D₈, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄×C₈, D₄⋊C₄, C₂.D₈, C₂.D₈, C₂.D₈, C₂×C₁₆, C₄×D₄, C₄⋊Q₈, C₂×D₈, C₂.D₁₆, C₄⋊C₁₆, C₁₆⋊₄C₄, C₄×D₈, C₈⋊₂Q₈, D₈⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, SD₃₂, C₂²⋊Q₈, C₂×D₈, C₈.C₂², D₄⋊Q₈, C₂×SD₃₂, C₁₆⋊C₂², D₈⋊Q₈

Character table of D₈⋊Q₈

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	8	8	2	2	2	2	4	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	0	0	2	-2	-2	2	-2	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	2	-2	-2	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₁	2	2	2	2	0	0	2	2	2	2	2	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	-√2	√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₃	2	2	2	2	0	0	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	√2	-√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	0	0	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	-2	2	2	-2	2	0	0	-2	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-2	-2	2	-2	2	2	0	0	-2	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	0	0	2	0	0	-2	0	-2i	2i	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	0	0	2	0	0	-2	0	2i	-2i	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₀	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₁	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₂	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₃	2	-2	2	-2	0	0	0	2	-2	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₄	2	-2	2	-2	0	0	0	2	-2	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₅	2	-2	2	-2	0	0	0	2	-2	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₆	2	-2	2	-2	0	0	0	2	-2	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₇	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
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	-4	0	0	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2

Smallest permutation representation of D₈⋊Q₈
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

Matrix representation of D₈⋊Q₈ ►in GL₄(𝔽₁₇) generated by

3	14	0	0
3	3	0	0
0	0	1	0
0	0	0	1

3	14	0	0
14	14	0	0
0	0	1	0
0	0	0	1

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

10	16	0	0
16	7	0	0
0	0	10	10
0	0	12	7

G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,1,16,0,0,2,16],[10,16,0,0,16,7,0,0,0,0,10,12,0,0,10,7] >;

D₈⋊Q₈ in GAP, Magma, Sage, TeX

D_8\rtimes Q_8

% in TeX

G:=Group("D8:Q8");

// GroupNames label

G:=SmallGroup(128,958);

// by ID

G=gap.SmallGroup(128,958);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,504,141,64,422,1684,438,242,4037,1027,124]);

// Polycyclic

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

// generators/relations

Export

Character table of D₈⋊Q₈ in TeX

G = D8⋊Q8 order 128 = 27

2nd semidirect product of D8 and Q8 acting via Q8/C4=C2

G = D₈⋊Q₈ order 128 = 2⁷

2^nd semidirect product of D₈ and Q₈ acting via Q₈/C₄=C₂