D8:1Q8 - GroupNames

G = D₈⋊₁Q₈ order 128 = 2⁷

1^st semidirect product of D₈ and Q₈ acting via Q₈/C₄=C₂

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

Aliases: D₈⋊₁Q₈, C₄.₁₃D₁₆, C₄².₁₄₅D₄, C₄⋊C₁₆⋊₆C₂, C₈⋊₂Q₈⋊₉C₂, (C₄×D₈).₆C₂, C₁₆⋊₃C₄⋊₄C₂, C₂.₇(C₂×D₁₆), (C₂×C₈).₆₉D₄, C₈.₂₉(C₂×Q₈), (C₂×C₄).₁₅₁D₈, C₂.D₁₆.₃C₂, C₈.₆₃(C₄○D₄), (C₄×C₈).₆₅C₂², (C₂×C₁₆).₇C₂², (C₂×C₈).₅₂₆C₂³, C₂².₁₁₂(C₂×D₈), C₄.₈(C₈.C₂²), C₄.₄₅(C₂²⋊Q₈), C₂.D₈.₁₁C₂², C₂.₁₂(D₄⋊Q₈), C₂.₁₄(Q₃₂⋊C₂), (C₂×D₈).₁₁₂C₂², (C₂×C₄).₇₉₄(C₂×D₄), SmallGroup(128,956)

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

Derived series

C₁ — C₂×C₈ — D₈⋊₁Q₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×D₈ — C₄×D₈ — D₈⋊₁Q₈

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — D₈⋊₁Q₈

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — D₈⋊₁Q₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — D₈⋊₁Q₈

Generators and relations for D₈⋊₁Q₈
G = < a,b,c,d | a⁸=b²=c⁴=1, d²=c², bab=dad^-1=a^-1, ac=ca, bc=cb, 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⋊₃C₄, C₄×D₈, C₈⋊₂Q₈, D₈⋊₁Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₈, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₆, C₂²⋊Q₈, C₂×D₈, C₈.C₂², D₄⋊Q₈, C₂×D₁₆, Q₃₂⋊C₂, D₈⋊₁Q₈

Character table of D₈⋊₁Q₈

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	0	-2	2	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	orthogonal lifted from D₁₆
ρ₁₅	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
ρ₁₆	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	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	orthogonal lifted from D₁₆
ρ₁₇	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	orthogonal lifted from D₁₆
ρ₁₈	2	-2	2	-2	0	0	0	-2	2	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	orthogonal lifted from D₁₆
ρ₁₉	2	-2	2	-2	0	0	0	2	-2	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	orthogonal lifted from D₁₆
ρ₂₀	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	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	orthogonal lifted from D₁₆
ρ₂₁	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	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	orthogonal lifted from D₁₆
ρ₂₂	2	-2	2	-2	0	0	0	2	-2	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	orthogonal lifted from D₁₆
ρ₂₃	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
ρ₂₄	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
ρ₂₅	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
ρ₂₆	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
ρ₂₇	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
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 D₈⋊₁Q₈
►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 27 11 19)(2 28 12 20)(3 29 13 21)(4 30 14 22)(5 31 15 23)(6 32 16 24)(7 25 9 17)(8 26 10 18)(33 49 41 57)(34 50 42 58)(35 51 43 59)(36 52 44 60)(37 53 45 61)(38 54 46 62)(39 55 47 63)(40 56 48 64)

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

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

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

Matrix representation of D₈⋊₁Q₈ ►in GL₄(𝔽₁₇) generated by

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

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

0	1	0	0
16	0	0	0
0	0	1	0
0	0	0	1

7	1	0	0
1	10	0	0
0	0	4	6
0	0	6	13

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,14,14,0,0,3,14],[1,0,0,0,0,1,0,0,0,0,14,3,0,0,3,3],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[7,1,0,0,1,10,0,0,0,0,4,6,0,0,6,13] >;

D₈⋊₁Q₈ in GAP, Magma, Sage, TeX

D_8\rtimes_1Q_8

% in TeX

G:=Group("D8:1Q8");

// GroupNames label

G:=SmallGroup(128,956);

// by ID

G=gap.SmallGroup(128,956);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,1684,438,242,4037,1027,124]);

// 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,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;

// generators/relations

Export

Character table of D₈⋊₁Q₈ in TeX

G = D8⋊1Q8 order 128 = 27

1st semidirect product of D8 and Q8 acting via Q8/C4=C2

G = D₈⋊₁Q₈ order 128 = 2⁷

1^st semidirect product of D₈ and Q₈ acting via Q₈/C₄=C₂