D8:8D4 - GroupNames

G = D₈⋊₈D₄ order 128 = 2⁷

2^nd semidirect product of D₈ and D₄ acting via D₄/C₂²=C₂

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

Aliases: D₈⋊₈D₄, Q₁₆⋊₈D₄, C₂³.₁₄D₈, (C₂×D₁₆)⋊₂C₂, C₈.₆₂(C₂×D₄), C₂.D₁₆⋊₉C₂, C₈⋊₇D₄⋊₁₆C₂, C₂²⋊C₁₆⋊₇C₂, (C₂×SD₃₂)⋊₈C₂, (C₂×C₄).₁₁₂D₈, (C₂×C₈).₁₇₂D₄, C₂.Q₃₂⋊₄C₂, C₄.₁₈C₂²≀C₂, C₂.₆(C₄○D₁₆), (C₂×C₁₆).₁C₂², C₂².₉₆(C₂×D₈), C₄.₁₀(C₈⋊C₂²), C₂.₇(C₁₆⋊C₂²), (C₂×C₈).₅₁₀C₂³, C₂.D₈.₂C₂², (C₂²×C₄).₃₄₅D₄, C₂.₂₆(C₂²⋊D₈), (C₂×D₈).₁₀₇C₂², (C₂²×C₈).₁₇₂C₂², (C₂×Q₁₆).₁₀₇C₂², (C₂×C₄○D₈)⋊₁C₂, (C₂×C₄).₇₇₈(C₂×D₄), SmallGroup(128,918)

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

Derived series

C₁ — C₂×C₈ — D₈⋊₈D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂×C₄○D₈ — D₈⋊₈D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — D₈⋊₈D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — D₈⋊₈D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — D₈⋊₈D₄

Generators and relations for D₈⋊₈D₄
G = < a,b,c,d | a⁸=b²=c⁴=d²=1, bab=cac^-1=dad=a^-1, cbc^-1=ab, dbd=a⁵b, dcd=c^-1 >

Subgroups: 324 in 115 conjugacy classes, 34 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₁₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, D₈, SD₁₆, Q₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, D₄⋊C₄, C₂.D₈, C₂×C₁₆, D₁₆, SD₃₂, C₄⋊D₄, C₂²×C₈, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₂×C₄○D₄, C₂²⋊C₁₆, C₂.D₁₆, C₂.Q₃₂, C₈⋊₇D₄, C₂×D₁₆, C₂×SD₃₂, C₂×C₄○D₈, D₈⋊₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₂²≀C₂, C₂×D₈, C₈⋊C₂², C₂²⋊D₈, C₄○D₁₆, C₁₆⋊C₂², D₈⋊₈D₄

Character table of D₈⋊₈D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	4	8	8	16	2	2	2	2	8	8	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	-2	2	0	2	0	0	-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	0	0	-2	2	-2	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	0	0	2	0	0	-2	-2	2	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	2	-2	0	2	0	0	-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	0	0	0	2	-2	-2	2	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	2	0	0	0	2	2	2	2	0	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	0	0	0	-2	2	2	-2	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	0	0	0	-2	2	2	-2	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	0	0	0	-2	-2	-2	-2	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	0	0	0	-2	-2	-2	-2	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	0	0	0	0	0	2i	-2i	0	0	0	0	√2	√2	-√2	-√2	-√-2	√-2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from C₄○D₁₆
ρ₂₀	2	-2	2	-2	0	0	0	0	0	-2i	2i	0	0	0	0	√2	√2	-√2	-√2	√-2	-√-2	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₁	2	-2	2	-2	0	0	0	0	0	2i	-2i	0	0	0	0	-√2	-√2	√2	√2	√-2	-√-2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from C₄○D₁₆
ρ₂₂	2	-2	2	-2	0	0	0	0	0	-2i	2i	0	0	0	0	-√2	-√2	√2	√2	-√-2	√-2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₃	2	-2	2	-2	0	0	0	0	0	-2i	2i	0	0	0	0	-√2	-√2	√2	√2	-√-2	√-2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from C₄○D₁₆
ρ₂₄	2	-2	2	-2	0	0	0	0	0	2i	-2i	0	0	0	0	√2	√2	-√2	-√2	-√-2	√-2	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₅	2	-2	2	-2	0	0	0	0	0	-2i	2i	0	0	0	0	√2	√2	-√2	-√2	√-2	-√-2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from C₄○D₁₆
ρ₂₆	2	-2	2	-2	0	0	0	0	0	2i	-2i	0	0	0	0	-√2	-√2	√2	√2	√-2	-√-2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₇	4	-4	-4	4	0	0	0	0	-4	0	0	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⋊₈D₄
►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 50)(2 49)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 26)(10 25)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 45)(18 44)(19 43)(20 42)(21 41)(22 48)(23 47)(24 46)(33 59)(34 58)(35 57)(36 64)(37 63)(38 62)(39 61)(40 60)

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

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

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

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

Matrix representation of D₈⋊₈D₄ ►in GL₄(𝔽₁₇) generated by

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

0	16	0	0
16	0	0	0
0	0	11	4
0	0	4	6

0	1	0	0
16	0	0	0
0	0	0	4
0	0	4	0

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

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

D₈⋊₈D₄ in GAP, Magma, Sage, TeX

D_8\rtimes_8D_4

% in TeX

G:=Group("D8:8D4");

// GroupNames label

G:=SmallGroup(128,918);

// by ID

G=gap.SmallGroup(128,918);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,352,1123,570,360,4037,2028,124]);

// Polycyclic

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

// generators/relations

Export

Character table of D₈⋊₈D₄ in TeX

G = D8⋊8D4 order 128 = 27

2nd semidirect product of D8 and D4 acting via D4/C22=C2

G = D₈⋊₈D₄ order 128 = 2⁷

2^nd semidirect product of D₈ and D₄ acting via D₄/C₂²=C₂