D4.1Q16 - GroupNames

G = D₄.₁Q₁₆ order 128 = 2⁷

1^st non-split extension by D₄ of Q₁₆ acting via Q₁₆/C₈=C₂

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

Aliases: D₄.₁Q₁₆, C₈.₂₆SD₁₆, C₄².₂₂₆C₂³, C₈⋊₁C₈⋊₇C₂, C₈⋊₂Q₈⋊₅C₂, (C₈×D₄).₆C₂, C₄⋊C₄.₂₀₁D₄, (C₂×C₈).₃₄₅D₄, C₄.₂₅(C₂×Q₁₆), (C₂×D₄).₁₉₁D₄, C₄.₄₂(C₄○D₈), (C₄×C₈).₅₅C₂², D₄⋊₂Q₈.₈C₂, C₄.₅₉(C₂×SD₁₆), C₄⋊Q₈.₄₉C₂², C₄⋊C₈.₁₇₈C₂², C₄.₆Q₁₆⋊₁₅C₂, C₂.₈(D₄.D₄), C₂.₈(D₄.₄D₄), C₂.₈(C₈.₁₈D₄), (C₄×D₄).₂₇₉C₂², C₄.₁₁₄(C₈.C₂²), C₂².₁₈₇(C₄⋊D₄), (C₂×C₄).₁₁(C₄○D₄), (C₂×C₄).₁₂₆₁(C₂×D₄), SmallGroup(128,407)

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

Derived series

C₁ — C₄² — D₄.₁Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×D₄ — C₈×D₄ — D₄.₁Q₁₆

Lower central

C₁ — C₂² — C₄² — D₄.₁Q₁₆

Upper central

C₁ — C₂² — C₄² — D₄.₁Q₁₆

Jennings

C₁ — C₂² — C₂² — C₄² — D₄.₁Q₁₆

Generators and relations for D₄.₁Q₁₆
G = < a,b,c,d | a⁴=b²=c⁸=1, d²=a²c⁴, bab=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=ab, dcd^-1=c^-1 >

Subgroups: 176 in 80 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄×C₈, C₂²⋊C₈, D₄⋊C₄, C₄⋊C₈, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₄×D₄, C₄⋊Q₈, C₂²×C₈, C₄.₆Q₁₆, C₈⋊₁C₈, C₈×D₄, D₄⋊₂Q₈, C₈⋊₂Q₈, D₄.₁Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, Q₁₆, C₂×D₄, C₄○D₄, C₄⋊D₄, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₈.C₂², D₄.D₄, C₈.₁₈D₄, D₄.₄D₄, D₄.₁Q₁₆

Character table of D₄.₁Q₁₆

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N
size	1	1	1	1	4	4	2	2	2	2	4	4	4	16	16	2	2	2	2	4	4	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	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	-2	-2	2	-2	2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	-2	2	-2	0	-2	0	0	0	2	2	2	2	0	0	0	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	0	0	2	-2	2	-2	0	-2	0	0	0	-2	-2	-2	-2	0	0	0	2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	2	2	-2	2	-2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	-2	2	-2	0	2	0	-2	0	0	0	0	0	√2	-√2	-√2	√2	-√2	√2	√2	-√2	√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₄	2	2	-2	-2	-2	2	0	2	0	-2	0	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	-√2	√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₅	2	2	-2	-2	2	-2	0	2	0	-2	0	0	0	0	0	-√2	√2	√2	-√2	√2	-√2	-√2	√2	-√2	√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	2	-2	-2	-2	2	0	2	0	-2	0	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	√2	√2	-√2	-√2	0	0	0	0	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	2	2	2	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	2	-2	-2	0	0	0	-2	0	2	-2i	0	2i	0	0	√2	-√2	-√2	√2	-√-2	√-2	-√-2	√2	-√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₁₉	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₂₀	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₁	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	2	-2	2	-2	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₂₂	2	2	2	2	0	0	-2	-2	-2	-2	0	2	0	0	0	0	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	2	-2	-2	0	0	0	-2	0	2	2i	0	-2i	0	0	-√2	√2	√2	-√2	-√-2	√-2	-√-2	-√2	√2	√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₄	2	2	-2	-2	0	0	0	-2	0	2	2i	0	-2i	0	0	√2	-√2	-√2	√2	√-2	-√-2	√-2	√2	-√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₅	2	-2	2	-2	0	0	-2	0	2	0	0	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₆	2	2	-2	-2	0	0	0	-2	0	2	-2i	0	2i	0	0	-√2	√2	√2	-√2	√-2	-√-2	√-2	-√2	√2	-√-2	0	0	0	0	complex lifted from C₄○D₈
ρ₂₇	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 D₄.₄D₄
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 D₄.₄D₄
ρ₂₉	4	-4	4	-4	0	0	4	0	-4	0	0	0	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

Smallest permutation representation of D₄.₁Q₁₆
►On 64 points

Generators in S₆₄

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

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

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

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

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

Matrix representation of D₄.₁Q₁₆ ►in GL₄(𝔽₁₇) generated by

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

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

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

0	7	0	0
12	0	0	0
0	0	7	1
0	0	1	10

G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,15,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,3,3,0,0,14,3],[0,12,0,0,7,0,0,0,0,0,7,1,0,0,1,10] >;

D₄.₁Q₁₆ in GAP, Magma, Sage, TeX

D_4._1Q_{16}

% in TeX

G:=Group("D4.1Q16");

// GroupNames label

G:=SmallGroup(128,407);

// by ID

G=gap.SmallGroup(128,407);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,1123,136,2804,718,172]);

// Polycyclic

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

// generators/relations

Export

Character table of D₄.₁Q₁₆ in TeX

G = D4.1Q16 order 128 = 27

1st non-split extension by D4 of Q16 acting via Q16/C8=C2

G = D₄.₁Q₁₆ order 128 = 2⁷

1^st non-split extension by D₄ of Q₁₆ acting via Q₁₆/C₈=C₂