Q16.5D4 - GroupNames

G = Q₁₆.₅D₄ order 128 = 2⁷

2^nd non-split extension by Q₁₆ of D₄ acting via D₄/C₄=C₂

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

Aliases: Q₁₆.₅D₄, C₄².₁₄₁D₄, C₄⋊C₁₆⋊₈C₂, (C₂×Q₃₂)⋊₄C₂, (C₂×C₄).₃₅D₈, C₈.₇₅(C₂×D₄), (C₄×Q₁₆)⋊₁₉C₂, (C₂×C₈).₁₇₅D₄, C₂.D₁₆.₂C₂, C₂.₉(C₄○D₁₆), C₈.₉₃(C₄○D₄), (C₂×C₁₆).₆C₂², (C₂×SD₃₂).₄C₂, (C₂×D₈).₄C₂², C₂.Q₃₂⋊₁₁C₂, C₂.₂₃(C₄⋊D₈), C₄.₅₄(C₄⋊D₄), C₄.₁₉(C₈⋊C₂²), (C₄×C₈).₁₀₄C₂², (C₂×C₈).₅₁₉C₂³, C₈.₁₂D₄.₂C₂, C₂².₁₀₅(C₂×D₈), (C₂×Q₁₆).₅C₂², C₂.₁₁(Q₃₂⋊C₂), C₂.D₈.₁₆₁C₂², (C₂×C₄).₇₈₇(C₂×D₄), SmallGroup(128,943)

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

Derived series

C₁ — C₂×C₈ — Q₁₆.₅D₄

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×Q₁₆ — C₄×Q₁₆ — Q₁₆.₅D₄

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — Q₁₆.₅D₄

Upper central

C₁ — C₂² — C₄² — C₄×C₈ — Q₁₆.₅D₄

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — Q₁₆.₅D₄

Generators and relations for Q₁₆.₅D₄
G = < a,b,c,d | a⁸=c⁴=1, b²=d²=a⁴, bab^-1=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=a^-1b, dcd^-1=a⁴c^-1 >

Subgroups: 204 in 79 conjugacy classes, 32 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₁₆, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄×C₈, Q₈⋊C₄, C₂.D₈, C₂×C₁₆, SD₃₂, Q₃₂, C₄×Q₈, C₄.₄D₄, C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₂.D₁₆, C₂.Q₃₂, C₄⋊C₁₆, C₄×Q₁₆, C₈.₁₂D₄, C₂×SD₃₂, C₂×Q₃₂, Q₁₆.₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×D₈, C₈⋊C₂², C₄⋊D₈, C₄○D₁₆, Q₃₂⋊C₂, Q₁₆.₅D₄

Character table of Q₁₆.₅D₄

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	16	2	2	2	2	4	8	8	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	0	0	-2	0	2	0	-2	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	2	-2	-2	2	-2	0	0	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	2	0	0	-2	0	-2	0	2	0	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	2	2	2	0	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	0	-2	-2	-2	-2	2	0	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	-2	2	2	-2	-2	0	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	-2	2	2	-2	-2	0	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	-2	-2	-2	-2	2	0	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	2	0	0	-2	0	0	-2i	0	2i	0	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	2	0	0	-2	0	0	2i	0	-2i	0	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	2i	-2i	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√-2	√-2	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₀	2	2	-2	-2	0	0	-2i	2i	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√-2	-√-2	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex lifted from C₄○D₁₆
ρ₂₁	2	2	-2	-2	0	0	-2i	2i	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√-2	-√-2	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from C₄○D₁₆
ρ₂₂	2	2	-2	-2	0	0	2i	-2i	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√-2	√-2	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from C₄○D₁₆
ρ₂₃	2	2	-2	-2	0	0	2i	-2i	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√-2	-√-2	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from C₄○D₁₆
ρ₂₄	2	2	-2	-2	0	0	2i	-2i	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√-2	-√-2	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₅	2	2	-2	-2	0	0	-2i	2i	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√-2	√-2	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex lifted from C₄○D₁₆
ρ₂₆	2	2	-2	-2	0	0	-2i	2i	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√-2	√-2	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from C₄○D₁₆
ρ₂₇	4	-4	-4	4	0	-4	0	0	4	0	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	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 Q₁₆.₅D₄
►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 47 15 35)(2 48 16 36)(3 41 9 37)(4 42 10 38)(5 43 11 39)(6 44 12 40)(7 45 13 33)(8 46 14 34)(17 61 29 49)(18 62 30 50)(19 63 31 51)(20 64 32 52)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)

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

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

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

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

Matrix representation of Q₁₆.₅D₄ ►in GL₄(𝔽₁₇) generated by

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

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

4	0	0	0
0	4	0	0
0	0	1	15
0	0	1	16

4	0	0	0
0	13	0	0
0	0	1	15
0	0	0	16

G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[1,7,0,0,7,16,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[4,0,0,0,0,13,0,0,0,0,1,0,0,0,15,16] >;

Q₁₆.₅D₄ in GAP, Magma, Sage, TeX

Q_{16}._5D_4

% in TeX

G:=Group("Q16.5D4");

// GroupNames label

G:=SmallGroup(128,943);

// by ID

G=gap.SmallGroup(128,943);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,512,422,352,1684,438,242,4037,1027,124]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₁₆.₅D₄ in TeX

G = Q16.5D4 order 128 = 27

2nd non-split extension by Q16 of D4 acting via D4/C4=C2

G = Q₁₆.₅D₄ order 128 = 2⁷

2^nd non-split extension by Q₁₆ of D₄ acting via D₄/C₄=C₂