D8.10D4 - GroupNames

G = D₈.₁₀D₄ order 128 = 2⁷

2^nd non-split extension by D₈ of D₄ acting via D₄/C₂²=C₂

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

Aliases: D₈.₁₀D₄, Q₁₆.₉D₄, C₂³.₁₅D₈, (C₂×Q₃₂)⋊₂C₂, C₈.₆₅(C₂×D₄), C₂.D₁₆⋊₅C₂, C₂²⋊C₁₆⋊₈C₂, (C₂×C₈).₁₇₃D₄, (C₂×C₄).₁₁₃D₈, C₂.Q₃₂⋊₉C₂, C₄.₂₁C₂²≀C₂, (C₂×SD₃₂)⋊₁₀C₂, C₂.₇(C₄○D₁₆), C₂².₉₉(C₂×D₈), C₈.₁₈D₄⋊₁₆C₂, C₄.₁₃(C₈⋊C₂²), C₂.D₈.₄C₂², (C₂×C₈).₅₁₃C₂³, (C₂×C₁₆).₃₇C₂², C₂.₈(Q₃₂⋊C₂), (C₂²×C₄).₃₄₈D₄, C₂.₂₉(C₂²⋊D₈), (C₂×D₈).₁₀₉C₂², (C₂²×C₈).₁₇₃C₂², (C₂×Q₁₆).₁₀₈C₂², (C₂×C₄○D₈).₅C₂, (C₂×C₄).₇₈₁(C₂×D₄), SmallGroup(128,921)

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

Derived series

C₁ — C₂×C₈ — D₈.₁₀D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂²×C₈ — C₂×C₄○D₈ — D₈.₁₀D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — D₈.₁₀D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — D₈.₁₀D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — D₈.₁₀D₄

Generators and relations for D₈.₁₀D₄
G = < a,b,c,d | a⁸=b²=c⁴=1, d²=a⁴, bab=cac^-1=dad^-1=a^-1, cbc^-1=dbd^-1=a⁵b, dcd^-1=c^-1 >

Subgroups: 276 in 110 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, Q₈⋊C₄, C₂.D₈, C₂×C₁₆, SD₃₂, Q₃₂, C₂²⋊Q₈, C₂²×C₈, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₂×C₄○D₄, C₂²⋊C₁₆, C₂.D₁₆, C₂.Q₃₂, C₈.₁₈D₄, C₂×SD₃₂, C₂×Q₃₂, C₂×C₄○D₈, D₈.₁₀D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₂²≀C₂, C₂×D₈, C₈⋊C₂², C₂²⋊D₈, C₄○D₁₆, Q₃₂⋊C₂, D₈.₁₀D₄

Character table of D₈.₁₀D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	4	8	8	2	2	2	2	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	0	0	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	0	2	0	0	-2	2	-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	2	0	0	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	-2	2	2	0	0	-2	0	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	0	0	0	2	0	0	-2	-2	2	0	0	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	-2	-2	2	0	2	-2	2	0	0	-2	0	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	0	0	-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	2	0	0	-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
ρ₁₇	2	2	2	2	-2	0	0	-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
ρ₁₈	2	2	2	2	-2	0	0	-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
ρ₁₉	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	-√2	√2	√2	-√2	√-2	-√-2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₀	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	-√2	√2	√2	-√2	-√-2	√-2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₁	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	-√2	√2	√2	-√2	√-2	-√-2	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₂	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	-√2	√2	√2	-√2	-√-2	√-2	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₃	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	√2	-√2	-√2	√2	√-2	-√-2	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₄	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	√2	-√2	-√2	√2	-√-2	√-2	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₅	2	-2	2	-2	0	0	0	0	-2i	2i	0	0	0	0	0	√2	-√2	-√2	√2	√-2	-√-2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷-ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₆	2	-2	2	-2	0	0	0	0	2i	-2i	0	0	0	0	0	√2	-√2	-√2	√2	-√-2	√-2	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₇	4	-4	-4	4	0	0	0	-4	0	0	4	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 D₈.₁₀D₄
►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 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 30)(26 29)(27 28)(31 32)(33 40)(34 39)(35 38)(36 37)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(57 61)(58 60)(62 64)

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

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

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

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

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

Matrix representation of D₈.₁₀D₄ ►in GL₄(𝔽₁₇) generated by

16	0	0	0
0	16	0	0
0	0	0	11
0	0	3	11

16	0	0	0
0	1	0	0
0	0	0	11
0	0	14	0

0	1	0	0
16	0	0	0
0	0	10	5
0	0	4	7

0	1	0	0
1	0	0	0
0	0	11	14
0	0	1	6

G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,1,0,0,0,0,0,14,0,0,11,0],[0,16,0,0,1,0,0,0,0,0,10,4,0,0,5,7],[0,1,0,0,1,0,0,0,0,0,11,1,0,0,14,6] >;

D₈.₁₀D₄ in GAP, Magma, Sage, TeX

D_8._{10}D_4

% in TeX

G:=Group("D8.10D4");

// GroupNames label

G:=SmallGroup(128,921);

// by ID

G=gap.SmallGroup(128,921);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,352,1123,570,360,4037,2028,124]);

// Polycyclic

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

// generators/relations

Export

Character table of D₈.₁₀D₄ in TeX

G = D8.10D4 order 128 = 27

2nd non-split extension by D8 of D4 acting via D4/C22=C2

G = D₈.₁₀D₄ order 128 = 2⁷

2^nd non-split extension by D₈ of D₄ acting via D₄/C₂²=C₂