Q8:3D8 - GroupNames

G = Q₈⋊₃D₈ order 128 = 2⁷

2^nd semidirect product of Q₈ and D₈ acting via D₈/D₄=C₂

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q₈⋊₃D₈, D₄⋊₆SD₁₆, C₄².₁₈₈C₂³, Q₈⋊C₈⋊₈C₂, (D₄×Q₈)⋊₁C₂, D₄⋊C₈⋊₁₆C₂, C₄⋊C₄.₂₄D₄, C₄.₂₇(C₂×D₈), C₄⋊D₈.₂C₂, C₄.₄D₈⋊₂C₂, C₄⋊SD₁₆⋊₃₂C₂, (C₂×D₄).₂₅₀D₄, C₄.₁₀D₈⋊₈C₂, (C₄×C₈).₁₆C₂², (C₂×Q₈).₁₉₅D₄, C₄.₂₆(C₂×SD₁₆), C₄⋊Q₈.₁₀C₂², C₄⋊C₈.₁₆₁C₂², C₄.₅₉(C₈⋊C₂²), (C₄×D₄).₂₂C₂², (C₄×Q₈).₂₂C₂², C₂.₁₃(C₂²⋊D₈), C₂.₁₃(Q₈⋊D₄), C₄⋊₁D₄.₁₂C₂², C₄.₅₈(C₈.C₂²), C₂.₁₄(D₄.₈D₄), C₂².₁₅₄C₂²≀C₂, (C₂×C₄).₉₄₅(C₂×D₄), SmallGroup(128,359)

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

Derived series

C₁ — C₄² — Q₈⋊₃D₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×Q₈ — D₄×Q₈ — Q₈⋊₃D₈

Lower central

C₁ — C₂² — C₄² — Q₈⋊₃D₈

Upper central

C₁ — C₂² — C₄² — Q₈⋊₃D₈

Jennings

C₁ — C₂² — C₂² — C₄² — Q₈⋊₃D₈

Generators and relations for Q₈⋊₃D₈
G = < a,b,c,d | a⁴=c⁸=d²=1, b²=a², bab^-1=cac^-1=dad=a^-1, cbc^-1=dbd=a^-1b, dcd=c^-1 >

Subgroups: 320 in 125 conjugacy classes, 38 normal (32 characteristic)
C₁, C₂^[×3], C₂^[×3], C₄^[×4], C₄^[×7], C₂², C₂²^[×7], C₈^[×3], C₂×C₄^[×3], C₂×C₄^[×11], D₄^[×2], D₄^[×7], Q₈^[×2], Q₈^[×7], C₂³^[×2], C₄², C₄², C₂²⋊C₄^[×3], C₄⋊C₄^[×2], C₄⋊C₄^[×2], C₄⋊C₄^[×4], C₂×C₈^[×3], D₈^[×2], SD₁₆^[×2], C₂²×C₄^[×3], C₂×D₄, C₂×D₄^[×3], C₂×Q₈, C₂×Q₈^[×7], C₄×C₈, D₄⋊C₄^[×4], C₄⋊C₈^[×2], C₄×D₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈^[×3], C₄⋊₁D₄, C₄⋊Q₈, C₄⋊Q₈, C₂×D₈, C₂×SD₁₆, C₂²×Q₈, D₄⋊C₈, Q₈⋊C₈, C₄.₁₀D₈, C₄⋊D₈, C₄⋊SD₁₆, C₄.₄D₈, D₄×Q₈, Q₈⋊₃D₈
Quotients: C₁, C₂^[×7], C₂²^[×7], D₄^[×6], C₂³, D₈^[×2], SD₁₆^[×2], C₂×D₄^[×3], C₂²≀C₂, C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₈.C₂², C₂²⋊D₈, Q₈⋊D₄, D₄.₈D₄, Q₈⋊₃D₈

Character table of Q₈⋊₃D₈

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	4	4	16	2	2	2	2	4	4	4	8	8	8	8	4	4	4	4	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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	0	0	0	-2	-2	-2	-2	2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	0	-2	-2	-2	-2	2	0	0	0	0	-2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	0	-2	-2	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	0	0	0	-2	-2	2	2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	0	2	2	-2	-2	-2	0	0	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	2	0	2	2	-2	-2	-2	0	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	0	0	0	0	0	-2	2	0	-2	2	0	0	0	0	√2	√2	-√2	-√2	0	-√2	√2	0	orthogonal lifted from D₈
ρ₁₆	2	2	-2	-2	0	0	0	0	0	-2	2	0	-2	2	0	0	0	0	-√2	-√2	√2	√2	0	√2	-√2	0	orthogonal lifted from D₈
ρ₁₇	2	2	-2	-2	0	0	0	0	0	-2	2	0	2	-2	0	0	0	0	√2	√2	-√2	-√2	0	√2	-√2	0	orthogonal lifted from D₈
ρ₁₈	2	2	-2	-2	0	0	0	0	0	-2	2	0	2	-2	0	0	0	0	-√2	-√2	√2	√2	0	-√2	√2	0	orthogonal lifted from D₈
ρ₁₉	2	-2	-2	2	2	-2	0	2	-2	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	-√-2	0	0	√-2	complex lifted from SD₁₆
ρ₂₀	2	-2	-2	2	-2	2	0	2	-2	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	-√-2	0	0	√-2	complex lifted from SD₁₆
ρ₂₁	2	-2	-2	2	2	-2	0	2	-2	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	√-2	0	0	-√-2	complex lifted from SD₁₆
ρ₂₂	2	-2	-2	2	-2	2	0	2	-2	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	√-2	0	0	-√-2	complex lifted from SD₁₆
ρ₂₃	4	4	-4	-4	0	0	0	0	0	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₄	4	-4	-4	4	0	0	0	-4	4	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
ρ₂₅	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	-2i	2i	0	0	0	0	complex lifted from D₄.₈D₄
ρ₂₆	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	2i	-2i	0	0	0	0	complex lifted from D₄.₈D₄

Smallest permutation representation of Q₈⋊₃D₈
►On 64 points

Generators in S₆₄

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

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

(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)

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

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

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

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

Matrix representation of Q₈⋊₃D₈ ►in GL₄(𝔽₁₇) generated by

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

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

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

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

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

Q₈⋊₃D₈ in GAP, Magma, Sage, TeX

Q_8\rtimes_3D_8

% in TeX

G:=Group("Q8:3D8");

// GroupNames label

G:=SmallGroup(128,359);

// by ID

G=gap.SmallGroup(128,359);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,456,422,352,1123,570,521,136,2804,1411,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₈⋊₃D₈ in TeX

G = Q8⋊3D8 order 128 = 27

2nd semidirect product of Q8 and D8 acting via D8/D4=C2

G = Q₈⋊₃D₈ order 128 = 2⁷

2^nd semidirect product of Q₈ and D₈ acting via D₈/D₄=C₂