Q16:2D4 - GroupNames

G = Q₁₆⋊₂D₄ order 128 = 2⁷

1^st semidirect product of Q₁₆ and D₄ acting via D₄/C₄=C₂

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

Aliases: Q₁₆⋊₂D₄, C₄⋊₃SD₃₂, C₄².₁₃₇D₄, C₄⋊C₁₆⋊₁₀C₂, (C₄×Q₁₆)⋊₂C₂, (C₂×C₈).₆₆D₄, C₈.₇₁(C₂×D₄), C₈⋊₄D₄.₈C₂, (C₂×C₄).₁₄₈D₈, C₂.D₁₆⋊₁₁C₂, C₂.₇(C₂×SD₃₂), (C₂×SD₃₂)⋊₁₃C₂, C₈.₈₉(C₄○D₄), (C₄×C₈).₆₂C₂², (C₂×D₈).₃C₂², C₄.₅₀(C₄⋊D₄), C₂.₁₉(C₄⋊D₈), C₄.₁₅(C₈⋊C₂²), (C₂×C₈).₅₁₅C₂³, (C₂×C₁₆).₃₉C₂², C₂².₁₀₁(C₂×D₈), C₂.₁₀(C₁₆⋊C₂²), C₂.D₈.₁₅₇C₂², (C₂×Q₁₆).₁₀₉C₂², (C₂×C₄).₇₈₃(C₂×D₄), SmallGroup(128,939)

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

Derived series

C₁ — C₂×C₈ — Q₁₆⋊₂D₄

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×Q₁₆ — C₄×Q₁₆ — Q₁₆⋊₂D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — Q₁₆⋊₂D₄

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — Q₁₆⋊₂D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — Q₁₆⋊₂D₄

Generators and relations for Q₁₆⋊₂D₄
G = < a,b,c,d | a⁸=c⁴=d²=1, b²=a⁴, bab^-1=dad=a^-1, ac=ca, bc=cb, dbd=a^-1b, dcd=c^-1 >

Subgroups: 268 in 87 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, C₄², C₄⋊C₄, C₂×C₈, D₈, Q₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₂.D₈, C₂×C₁₆, SD₃₂, C₄×Q₈, C₄⋊₁D₄, C₂×D₈, C₂×D₈, C₂×Q₁₆, C₂.D₁₆, C₄⋊C₁₆, C₄×Q₁₆, C₈⋊₄D₄, C₂×SD₃₂, Q₁₆⋊₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, SD₃₂, C₄⋊D₄, C₂×D₈, C₈⋊C₂², C₄⋊D₈, C₂×SD₃₂, C₁₆⋊C₂², Q₁₆⋊₂D₄

Character table of Q₁₆⋊₂D₄

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	16	16	2	2	2	2	4	8	8	8	8	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	0	0	-2	0	-2	0	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	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	0	0	-2	0	2	0	-2	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	-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	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	-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
ρ₁₇	2	-2	-2	2	0	0	2	0	0	-2	0	0	2i	0	-2i	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	0	-2i	0	2i	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	-4	0	0	4	0	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	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 Q₁₆⋊₂D₄
►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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 19 13 23)(10 18 14 22)(11 17 15 21)(12 24 16 20)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 51 45 55)(42 50 46 54)(43 49 47 53)(44 56 48 52)

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

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

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

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

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

Matrix representation of Q₁₆⋊₂D₄ ►in GL₄(𝔽₁₇) generated by

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

0	16	0	0
16	0	0	0
0	0	10	1
0	0	1	7

0	13	0	0
13	0	0	0
0	0	16	0
0	0	0	16

0	13	0	0
4	0	0	0
0	0	16	0
0	0	0	1

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

Q₁₆⋊₂D₄ in GAP, Magma, Sage, TeX

Q_{16}\rtimes_2D_4

% in TeX

G:=Group("Q16:2D4");

// GroupNames label

G:=SmallGroup(128,939);

// by ID

G=gap.SmallGroup(128,939);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,352,1684,438,242,4037,1027,124]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₁₆⋊₂D₄ in TeX

G = Q16⋊2D4 order 128 = 27

1st semidirect product of Q16 and D4 acting via D4/C4=C2

G = Q₁₆⋊₂D₄ order 128 = 2⁷

1^st semidirect product of Q₁₆ and D₄ acting via D₄/C₄=C₂