C2^3.Q16 - GroupNames

G = C₂³.Q₁₆ order 128 = 2⁷

1^st non-split extension by C₂³ of Q₁₆ acting via Q₁₆/C₂=D₄

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

Aliases: C₂⁴.₃D₄, C₂³.₁Q₁₆, C₂³.₅SD₁₆, C₂.₆C₂≀C₄, C₂³⋊C₈.₄C₂, (C₂²×Q₈)⋊₁C₄, C₂².₁₇C₄≀C₂, (C₂²×C₄).₁₀D₄, C₂.C₄²⋊₅C₄, C₂³⋊Q₈.₁C₂, C₂.₅(C₄²⋊₃C₄), C₂³.₉D₄.₄C₂, C₂².₅₈(C₂³⋊C₄), C₂³.₁₅₈(C₂²⋊C₄), C₂².₁₆(Q₈⋊C₄), C₂.₄(C₂³.₃₁D₄), (C₂²×C₄).₅(C₂×C₄), (C₂×C₂²⋊C₄).₈₄C₂², SmallGroup(128,83)

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

Derived series

C₁ — C₂²×C₄ — C₂³.Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂×C₂²⋊C₄ — C₂³⋊Q₈ — C₂³.Q₁₆

Lower central

C₁ — C₂ — C₂³ — C₂²×C₄ — C₂³.Q₁₆

Upper central

C₁ — C₂² — C₂³ — C₂×C₂²⋊C₄ — C₂³.Q₁₆

Jennings

C₁ — C₂² — C₂³ — C₂×C₂²⋊C₄ — C₂³.Q₁₆

Generators and relations for C₂³.Q₁₆
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=bd⁴, dad^-1=ab=ba, ac=ca, eae^-1=abc, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=acd^-1 >

Subgroups: 252 in 81 conjugacy classes, 20 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, Q₈, C₂³, C₂³, C₂²⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²×Q₈, C₂³⋊C₈, C₂³.₉D₄, C₂³⋊Q₈, C₂³.Q₁₆
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, SD₁₆, Q₁₆, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, C₂³.₃₁D₄, C₂≀C₄, C₄²⋊₃C₄, C₂³.Q₁₆

Character table of C₂³.Q₁₆

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D
size	1	1	1	1	2	2	4	4	4	4	8	8	8	8	8	8	8	8	8	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	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	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	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	linear of order 2
ρ₅	1	1	1	1	1	1	-1	-1	-1	-1	-i	-1	i	1	1	i	-i	1	-1	-i	i	i	-i	linear of order 4
ρ₆	1	1	1	1	1	1	-1	-1	-1	-1	i	1	-i	-1	1	-i	i	-1	1	-i	i	i	-i	linear of order 4
ρ₇	1	1	1	1	1	1	-1	-1	-1	-1	-i	1	i	-1	1	i	-i	-1	1	i	-i	-i	i	linear of order 4
ρ₈	1	1	1	1	1	1	-1	-1	-1	-1	i	-1	-i	1	1	-i	i	1	-1	i	-i	-i	i	linear of order 4
ρ₉	2	2	2	2	2	2	2	2	-2	-2	0	0	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	-2	-2	2	2	0	0	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₂	2	-2	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₃	2	-2	2	-2	-2	2	0	0	2i	-2i	1-i	0	-1-i	0	0	1+i	-1+i	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	-2	2	-2	-2	2	0	0	-2i	2i	-1-i	0	1-i	0	0	-1+i	1+i	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₅	2	-2	2	-2	-2	2	0	0	2i	-2i	-1+i	0	1+i	0	0	-1-i	1-i	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₆	2	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	-2	2	-2	-2	2	0	0	-2i	2i	1+i	0	-1+i	0	0	1-i	-1-i	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₈	2	-2	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₉	4	4	-4	-4	0	0	0	0	0	0	0	-2	0	0	0	0	0	0	2	0	0	0	0	orthogonal lifted from C₂≀C₄
ρ₂₀	4	4	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₁	4	4	-4	-4	0	0	0	0	0	0	0	2	0	0	0	0	0	0	-2	0	0	0	0	orthogonal lifted from C₂≀C₄
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	0	0	0	0	0	complex lifted from C₄²⋊₃C₄
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	0	0	-2i	0	0	0	2i	0	0	0	0	0	complex lifted from C₄²⋊₃C₄

Smallest permutation representation of C₂³.Q₁₆
►On 32 points

Generators in S₃₂

(1 5)(2 10)(3 18)(4 31)(6 14)(7 22)(8 27)(9 13)(11 26)(12 23)(15 30)(16 19)(17 25)(20 24)(21 29)(28 32)

(1 13)(2 25)(3 15)(4 27)(5 9)(6 29)(7 11)(8 31)(10 17)(12 19)(14 21)(16 23)(18 30)(20 32)(22 26)(24 28)

(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)

(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)

(1 6 9 25)(2 13 29 5)(3 16 11 19)(4 18 31 26)(7 12 15 23)(8 22 27 30)(10 24 21 32)(14 20 17 28)

G:=sub<Sym(32)| (1,5)(2,10)(3,18)(4,31)(6,14)(7,22)(8,27)(9,13)(11,26)(12,23)(15,30)(16,19)(17,25)(20,24)(21,29)(28,32), (1,13)(2,25)(3,15)(4,27)(5,9)(6,29)(7,11)(8,31)(10,17)(12,19)(14,21)(16,23)(18,30)(20,32)(22,26)(24,28), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (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), (1,6,9,25)(2,13,29,5)(3,16,11,19)(4,18,31,26)(7,12,15,23)(8,22,27,30)(10,24,21,32)(14,20,17,28)>;

G:=Group( (1,5)(2,10)(3,18)(4,31)(6,14)(7,22)(8,27)(9,13)(11,26)(12,23)(15,30)(16,19)(17,25)(20,24)(21,29)(28,32), (1,13)(2,25)(3,15)(4,27)(5,9)(6,29)(7,11)(8,31)(10,17)(12,19)(14,21)(16,23)(18,30)(20,32)(22,26)(24,28), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (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), (1,6,9,25)(2,13,29,5)(3,16,11,19)(4,18,31,26)(7,12,15,23)(8,22,27,30)(10,24,21,32)(14,20,17,28) );

G=PermutationGroup([[(1,5),(2,10),(3,18),(4,31),(6,14),(7,22),(8,27),(9,13),(11,26),(12,23),(15,30),(16,19),(17,25),(20,24),(21,29),(28,32)], [(1,13),(2,25),(3,15),(4,27),(5,9),(6,29),(7,11),(8,31),(10,17),(12,19),(14,21),(16,23),(18,30),(20,32),(22,26),(24,28)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(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)], [(1,6,9,25),(2,13,29,5),(3,16,11,19),(4,18,31,26),(7,12,15,23),(8,22,27,30),(10,24,21,32),(14,20,17,28)]])

Matrix representation of C₂³.Q₁₆ ►in GL₆(𝔽₁₇)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	2	0	0
0	0	0	16	0	0
0	0	0	0	1	2
0	0	0	0	0	16

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

14	3	0	0	0	0
14	14	0	0	0	0
0	0	1	1	0	16
0	0	16	16	0	0
0	0	0	16	16	16
0	0	0	0	1	1

12	12	0	0	0	0
12	5	0	0	0	0
0	0	0	1	1	1
0	0	0	16	0	0
0	0	16	16	0	16
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,1,16,0,0,0,0,1,16,16,0,0,0,0,0,16,1,0,0,16,0,16,1],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,0,0,16,0,0,0,1,16,16,0,0,0,1,0,0,0,0,0,1,0,16,1] >;

C₂³.Q₁₆ in GAP, Magma, Sage, TeX

C_2^3.Q_{16}

% in TeX

G:=Group("C2^3.Q16");

// GroupNames label

G:=SmallGroup(128,83);

// by ID

G=gap.SmallGroup(128,83);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,1690,521,2804]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.Q₁₆ in TeX

G = C23.Q16 order 128 = 27

1st non-split extension by C23 of Q16 acting via Q16/C2=D4

G = C₂³.Q₁₆ order 128 = 2⁷

1^st non-split extension by C₂³ of Q₁₆ acting via Q₁₆/C₂=D₄