C2^3.32D8 - GroupNames

G = C₂³.₃₂D₈ order 128 = 2⁷

3^rd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

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

Aliases: C₂³.₃₂D₈, C₂².₂Q₃₂, C₂².₂SD₃₂, C₂.D₈⋊₂C₄, C₄.₁₉C₄≀C₂, (C₂×Q₁₆)⋊₁C₄, (C₂×C₈).₃₀₁D₄, C₂²⋊C₁₆.₃C₂, C₄.₆(C₂³⋊C₄), (C₂×C₄).₁₄SD₁₆, C₂.₅(D₈⋊₂C₄), C₈.₁₈D₄.₁C₂, (C₂²×C₄).₁₈₆D₄, C₂.₃(C₂.Q₃₂), (C₂²×C₈).₉₈C₂², C₂².₄Q₁₆.₃₃C₂, C₂².₅₆(D₄⋊C₄), C₂.₁₃(C₂².SD₁₆), (C₂×C₈).₁₈(C₂×C₄), (C₂×C₄).₂₁₈(C₂²⋊C₄), SmallGroup(128,80)

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

Derived series

C₁ — C₂×C₈ — C₂³.₃₂D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₈.₁₈D₄ — C₂³.₃₂D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₂³.₃₂D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₃₂D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂²×C₈ — C₂³.₃₂D₈

Generators and relations for C₂³.₃₂D₈
G = < a,b,c,d,e | a²=b²=c²=1, d⁸=c, e²=a, ab=ba, ac=ca, dad^-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=ad⁷ >

C₁

C₂

₂C₂

C₂²

C₄

₂C₂²

₂C₄

₈C₄

C₂×C₄

C₂³

₂C₂×C₄

₂C₈

₂C₂×C₄

₂C₈

₄Q₈

₄C₂×C₄

₄Q₈

₄C₂×C₄

₈C₂×C₄

C₂×C₈

C₂²×C₄

C₂×C₈

₂C₄⋊C₄

₂C₂×Q₈

₂C₄⋊C₄

₄C₁₆

₄C₂²⋊C₄

₄Q₁₆

₄C₂×C₈

₄C₄⋊C₄

₄C₂²×C₄

C₂²×C₈

C₂×Q₁₆

C₂.D₈

₂Q₈⋊C₄

₂C₂×C₁₆

₂C₂×C₄⋊C₄

₂C₂²⋊Q₈

C₂²⋊C₁₆

C₈.₁₈D₄

C₂².₄Q₁₆

C₂³.₃₂D₈

Character table of C₂³.₃₂D₈

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	2	2	2	2	4	8	8	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	-i	i	-i	i	-1	1	-1	-1	-1	-1	1	1	i	-i	i	i	-i	-i	-i	i	linear of order 4
ρ₆	1	1	1	1	-1	-1	1	1	-1	-i	i	-i	i	1	-1	-1	-1	-1	-1	1	1	-i	i	-i	-i	i	i	i	-i	linear of order 4
ρ₇	1	1	1	1	-1	-1	1	1	-1	i	-i	i	-i	1	-1	-1	-1	-1	-1	1	1	i	-i	i	i	-i	-i	-i	i	linear of order 4
ρ₈	1	1	1	1	-1	-1	1	1	-1	i	-i	i	-i	-1	1	-1	-1	-1	-1	1	1	-i	i	-i	-i	i	i	i	-i	linear of order 4
ρ₉	2	2	2	2	-2	-2	2	2	-2	0	0	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	2	2	2	2	2	0	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	2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	-√2	-√2	-√2	√2	√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₂	2	2	2	2	2	2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	√2	√2	√2	-√2	-√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₃	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	symplectic lifted from Q₃₂, Schur index 2
ρ₁₄	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	symplectic lifted from Q₃₂, Schur index 2
ρ₁₅	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₁₆	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	symplectic lifted from Q₃₂, Schur index 2
ρ₁₇	2	-2	2	-2	0	0	-2	2	0	-1+i	1+i	1-i	-1-i	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₈	2	-2	2	-2	0	0	-2	2	0	1+i	-1+i	-1-i	1-i	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₁₉	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	√-2	-√-2	√-2	-√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₂₀	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	-√-2	√-2	-√-2	√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₂₁	2	-2	2	-2	0	0	-2	2	0	1-i	-1-i	-1+i	1+i	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₂₂	2	-2	2	-2	0	0	-2	2	0	-1-i	1-i	1+i	-1+i	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄≀C₂
ρ₂₃	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	complex lifted from SD₃₂
ρ₂₄	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₅	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	complex lifted from SD₃₂
ρ₂₆	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	complex lifted from SD₃₂
ρ₂₇	4	-4	4	-4	0	0	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
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	complex lifted from D₈⋊₂C₄
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	complex lifted from D₈⋊₂C₄

Smallest permutation representation of C₂³.₃₂D₈
►On 32 points

Generators in S₃₂

(2 29)(4 31)(6 17)(8 19)(10 21)(12 23)(14 25)(16 27)

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

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

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

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

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

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

Matrix representation of C₂³.₃₂D₈ ►in GL₄(𝔽₁₇) generated by

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

16	0	0	0
0	16	0	0
0	0	16	0
0	0	0	16

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

11	2	0	0
16	9	0	0
0	0	0	4
0	0	16	0

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

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[11,16,0,0,2,9,0,0,0,0,0,16,0,0,4,0],[16,0,0,0,15,1,0,0,0,0,16,0,0,0,0,4] >;

C₂³.₃₂D₈ in GAP, Magma, Sage, TeX

C_2^3._{32}D_8

% in TeX

G:=Group("C2^3.32D8");

// GroupNames label

G:=SmallGroup(128,80);

// by ID

G=gap.SmallGroup(128,80);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,2804,1411,172,4037,2028,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^7>;

// generators/relations

Export

Subgroup lattice of C₂³.₃₂D₈ in TeX
Character table of C₂³.₃₂D₈ in TeX

G = C23.32D8 order 128 = 27

3rd non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₃₂D₈ order 128 = 2⁷

3^rd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂