C2^3.31D8 - GroupNames

G = C₂³.₃₁D₈ order 128 = 2⁷

2^nd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂

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

Aliases: C₂³.₃₁D₈, C₂².₂D₁₆, C₂².₇SD₃₂, (C₂×D₈)⋊₁C₄, C₂.D₈⋊₁C₄, C₄.₁₈C₄≀C₂, C₂²⋊C₁₆⋊₃C₂, C₈⋊₇D₄.₁C₂, (C₂×C₈).₃₀₀D₄, C₄.₅(C₂³⋊C₄), (C₂×C₄).₁₃SD₁₆, C₂.₄(D₈⋊₂C₄), C₂.₃(C₂.D₁₆), (C₂²×C₄).₁₈₅D₄, C₂².₄Q₁₆⋊₃₂C₂, (C₂²×C₈).₉₇C₂², C₂².₅₅(D₄⋊C₄), C₂.₁₂(C₂².SD₁₆), (C₂×C₈).₁₇(C₂×C₄), (C₂×C₄).₂₁₇(C₂²⋊C₄), SmallGroup(128,79)

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

Derived series

C₁ — C₂×C₈ — C₂³.₃₁D₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₈⋊₇D₄ — C₂³.₃₁D₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₂³.₃₁D₈

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂³.₃₁D₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂²×C₈ — C₂³.₃₁D₈

Generators and relations for C₂³.₃₁D₈
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=acd⁷ >

Subgroups: 204 in 63 conjugacy classes, 22 normal (all characteristic)
C₁, C₂^[×3], C₂^[×3], C₄^[×2], C₄^[×4], C₂²^[×3], C₂²^[×5], C₈^[×2], C₂×C₄^[×2], C₂×C₄^[×7], D₄^[×4], C₂³, C₂³, C₁₆, C₂²⋊C₄, C₄⋊C₄^[×4], C₂×C₈^[×2], C₂×C₈, D₈, C₂²×C₄, C₂²×C₄, C₂×D₄^[×2], D₄⋊C₄, C₂.D₈, C₂×C₁₆, C₂×C₄⋊C₄, C₄⋊D₄, C₂²×C₈, C₂×D₈, C₂².₄Q₁₆, C₂²⋊C₁₆, C₈⋊₇D₄, C₂³.₃₁D₈
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], C₂²⋊C₄, D₈, SD₁₆, C₂³⋊C₄, D₄⋊C₄, C₄≀C₂, D₁₆, SD₃₂, C₂².SD₁₆, C₂.D₁₆, D₈⋊₂C₄, C₂³.₃₁D₈

Character table of C₂³.₃₁D₈

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	16	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	-i	i	-i	i	-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	-1	-i	i	-i	i	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	-1	i	-i	i	-i	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	-1	i	-i	i	-i	-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	0	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
ρ₁₀	2	2	2	2	2	2	0	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	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	orthogonal lifted from D₁₆
ρ₁₂	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	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	orthogonal lifted from D₁₆
ρ₁₃	2	2	2	2	2	2	0	-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	2	2	0	-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	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	orthogonal lifted from D₁₆
ρ₁₆	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	orthogonal lifted from D₁₆
ρ₁₇	2	-2	2	-2	0	0	0	2	-2	0	-1+i	1+i	1-i	-1-i	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	0	2	-2	0	-1-i	1-i	1+i	-1+i	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	0	-2	-2	2	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	0	-2	-2	2	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	0	2	-2	0	1+i	-1+i	-1-i	1-i	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	0	2	-2	0	1-i	-1-i	-1+i	1+i	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	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	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	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁷+ζ₁₆	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	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	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	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	ζ₁₆⁷+ζ₁₆	ζ₁₆⁷+ζ₁₆	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵+ζ₁₆³	complex lifted from SD₃₂
ρ₂₇	4	-4	4	-4	0	0	0	-4	4	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 C₂³.₃₁D₈
►On 32 points

Generators in S₃₂

(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 18)(16 20)

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

(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)

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

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

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

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

Matrix representation of C₂³.₃₁D₈ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	1	0
0	0	2	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

13	11	0	0
6	13	0	0
0	0	8	9
0	0	0	9

1	0	0	0
0	16	0	0
0	0	1	0
0	0	5	13

G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,2,0,0,0,16],[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],[13,6,0,0,11,13,0,0,0,0,8,0,0,0,9,9],[1,0,0,0,0,16,0,0,0,0,1,5,0,0,0,13] >;

C₂³.₃₁D₈ in GAP, Magma, Sage, TeX

C_2^3._{31}D_8

% in TeX

G:=Group("C2^3.31D8");

// GroupNames label

G:=SmallGroup(128,79);

// by ID

G=gap.SmallGroup(128,79);

# by ID

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

// 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*c*d^7>;

// generators/relations

G = C23.31D8 order 128 = 27

2nd non-split extension by C23 of D8 acting via D8/D4=C2

G = C₂³.₃₁D₈ order 128 = 2⁷

2^nd non-split extension by C₂³ of D₈ acting via D₈/D₄=C₂