C2.7C2wrC4 - GroupNames

G = C₂.₇C₂≀C₄ order 128 = 2⁷

4^th central stem extension by C₂ of C₂≀C₄

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

Aliases: C₂.₇C₂≀C₄, (C₂×C₄).₂Q₁₆, (C₂×C₄).₆SD₁₆, C₂².₂₀C₄≀C₂, (C₂²×C₄).₃₄D₄, (C₂²×Q₈).₃C₄, C₂.C₄².₆C₄, C₂².C₄².₉C₂, C₂².₆₁(C₂³⋊C₄), C₂.₅(C₄².₃C₄), C₂³.₁₆₁(C₂²⋊C₄), C₂².₁₉(Q₈⋊C₄), C₂.₇(C₂³.₃₁D₄), C₂³.₇₈C₂³.₁C₂, C₂².M₄(2).₆C₂, (C₂×C₄⋊C₄).₈C₂², (C₂²×C₄).₈(C₂×C₄), SmallGroup(128,86)

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

Derived series

C₁ — C₂²×C₄ — C₂.₇C₂≀C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇₈C₂³ — C₂.₇C₂≀C₄

Lower central

C₁ — C₂ — C₂³ — C₂²×C₄ — C₂.₇C₂≀C₄

Upper central

C₁ — C₂² — C₂³ — C₂×C₄⋊C₄ — C₂.₇C₂≀C₄

Jennings

C₁ — C₂² — C₂³ — C₂×C₄⋊C₄ — C₂.₇C₂≀C₄

Generators and relations for C₂.₇C₂≀C₄
G = < a,b,c,d,e,f | a²=d²=e²=1, b²=c²=f⁴=a, cbc^-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bd=db, be=eb, fbf^-1=bcde, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, ef=fe >

Subgroups: 180 in 70 conjugacy classes, 20 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂.C₄², C₂.C₄², C₂²⋊C₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×M₄(2), C₂²×Q₈, C₂².M₄(2), C₂².C₄², C₂³.₇₈C₂³, C₂.₇C₂≀C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, SD₁₆, Q₁₆, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, C₂³.₃₁D₄, C₂≀C₄, C₄².₃C₄, C₂.₇C₂≀C₄

Character table of C₂.₇C₂≀C₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H
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	-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	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	-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	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	-2	0	0	-2	0	0	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	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	√2	0	0	√2	0	0	-√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₂	2	-2	2	-2	2	-2	-2	0	2	0	0	0	0	0	0	-√2	0	0	-√2	0	0	√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₃	2	-2	2	-2	-2	2	0	2i	0	-2i	0	0	0	0	0	0	1+i	-1-i	0	1-i	-1+i	0	0	complex lifted from C₄≀C₂
ρ₁₄	2	-2	2	-2	-2	2	0	-2i	0	2i	0	0	0	0	0	0	1-i	-1+i	0	1+i	-1-i	0	0	complex lifted from C₄≀C₂
ρ₁₅	2	-2	2	-2	-2	2	0	-2i	0	2i	0	0	0	0	0	0	-1+i	1-i	0	-1-i	1+i	0	0	complex lifted from C₄≀C₂
ρ₁₆	2	-2	2	-2	2	-2	2	0	-2	0	0	0	0	0	0	-√-2	0	0	√-2	0	0	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	-2	2	-2	2	-2	2	0	-2	0	0	0	0	0	0	√-2	0	0	-√-2	0	0	-√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	-2	2	-2	-2	2	0	2i	0	-2i	0	0	0	0	0	0	-1-i	1+i	0	-1+i	1-i	0	0	complex lifted from C₄≀C₂
ρ₁₉	4	4	-4	-4	0	0	0	0	0	0	-2	0	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from C₂≀C₄
ρ₂₀	4	4	-4	-4	0	0	0	0	0	0	2	0	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from C₂≀C₄
ρ₂₁	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
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	2	0	-2	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	-2	0	2	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2

Smallest permutation representation of C₂.₇C₂≀C₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

(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)

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

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

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

Matrix representation of C₂.₇C₂≀C₄ ►in GL₆(𝔽₁₇)

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

0	3	0	0	0	0
11	0	0	0	0	0
0	0	0	16	0	0
0	0	16	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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

8	0	0	0	0	0
0	15	0	0	0	0
0	0	0	0	0	16
0	0	0	0	16	0
0	0	0	1	0	0
0	0	16	0	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,11,0,0,0,0,3,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0] >;

C₂.₇C₂≀C₄ in GAP, Magma, Sage, TeX

C_2._7C_2\wr C_4

% in TeX

G:=Group("C2.7C2wrC4");

// GroupNames label

G:=SmallGroup(128,86);

// by ID

G=gap.SmallGroup(128,86);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,1690,521,248,2804]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂.₇C₂≀C₄ in TeX

G = C2.7C2≀C4 order 128 = 27

4th central stem extension by C2 of C2≀C4

G = C₂.₇C₂≀C₄ order 128 = 2⁷

4^th central stem extension by C₂ of C₂≀C₄