C2^4.180C2^3 - GroupNames

G = C₂⁴.₁₈₀C₂³ order 128 = 2⁷

20^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₁₈₀C₂³, (C₂²×C₄)⋊₃Q₈, (C₂²×C₄).₈₀D₄, C₂³.₅₃(C₂×Q₈), C₂³.₅₈₈(C₂×D₄), C₂³⋊₂Q₈.₄C₂, (C₂³×C₄).₂₇C₂², C₂³.₉D₄.₇C₂, C₂².₂₁₄C₂²≀C₂, C₂³.₁₂₉(C₄○D₄), C₂.₁₁(C₂³⋊Q₈), C₂³.₃₄D₄.₆C₂, C₂².₃₄(C₂²⋊Q₈), C₂².₂₆(C₄.₄D₄), C₂.₁₇(C₂³.₇D₄), (C₂×C₂²⋊C₄).₁₈C₂², SmallGroup(128,762)

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

Derived series

C₁ — C₂⁴ — C₂⁴.₁₈₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₁₈₀C₂³

Lower central

C₁ — C₂ — C₂⁴ — C₂⁴.₁₈₀C₂³

Upper central

C₁ — C₂² — C₂⁴ — C₂⁴.₁₈₀C₂³

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₁₈₀C₂³

Generators and relations for C₂⁴.₁₈₀C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=f²=a, g²=d, ab=ba, ac=ca, ad=da, fef^-1=ae=ea, af=fa, ag=ga, bc=cb, fbf^-1=bd=db, geg^-1=be=eb, bg=gb, ece^-1=cd=dc, gfg^-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd >

Subgroups: 344 in 145 conjugacy classes, 42 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂³.₉D₄, C₂³.₃₄D₄, C₂³⋊₂Q₈, C₂⁴.₁₈₀C₂³
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂³⋊Q₈, C₂³.₇D₄, C₂⁴.₁₈₀C₂³

Character table of C₂⁴.₁₈₀C₂³

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	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	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	2	-2	2	-2	-2	-2	0	0	0	0	2	-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	0	0	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	2	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	0	2	0	0	0	0	0	0	0	-2	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	-2	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₅	2	-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	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-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	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-2i	0	0	0	2i	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	-2i	0	0	2i	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	-2	2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	2i	0	0	-2i	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	-2	2	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₄	4	-4	4	-4	0	0	0	0	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₅	4	4	-4	-4	0	0	0	0	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₆	4	-4	4	-4	0	0	0	0	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.₇D₄

Smallest permutation representation of C₂⁴.₁₈₀C₂³
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

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

Matrix representation of C₂⁴.₁₈₀C₂³ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	3	0	0
0	0	0	4	0	0
0	0	2	1	4	3
0	0	0	2	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	3	4	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	1	0	0	0	0
4	0	0	0	0	0
0	0	0	0	1	0
0	0	4	2	1	1
0	0	1	0	0	0
0	0	1	2	4	3

2	0	0	0	0	0
0	3	0	0	0	0
0	0	3	4	0	0
0	0	3	2	0	0
0	0	1	3	2	4
0	0	4	1	3	3

4	0	0	0	0	0
0	4	0	0	0	0
0	0	3	4	0	0
0	0	0	2	0	0
0	0	0	0	3	0
0	0	0	1	0	3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,2,0,0,0,3,4,1,2,0,0,0,0,4,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,3,0,0,0,4,0,4,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,1,0,0,0,2,0,2,0,0,1,1,0,4,0,0,0,1,0,3],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,3,1,4,0,0,4,2,3,1,0,0,0,0,2,3,0,0,0,0,4,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,4,2,0,1,0,0,0,0,3,0,0,0,0,0,0,3] >;

C₂⁴.₁₈₀C₂³ in GAP, Magma, Sage, TeX

C_2^4._{180}C_2^3

% in TeX

G:=Group("C2^4.180C2^3");

// GroupNames label

G:=SmallGroup(128,762);

// by ID

G=gap.SmallGroup(128,762);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,352,1411,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴.₁₈₀C₂³ in TeX

G = C24.180C23 order 128 = 27

20th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₁₈₀C₂³ order 128 = 2⁷

20^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²