C2^4.31D4 - GroupNames

G = C₂⁴.₃₁D₄ order 128 = 2⁷

31^st non-split extension by C₂⁴ of D₄ acting faithfully

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

Aliases: C₂⁴.₃₁D₄, C₂⁴.₁₇₈C₂³, C₂²⋊C₄⋊₆D₄, (C₂²×C₄)⋊₁₀D₄, C₂³.₅₈₄(C₂×D₄), C₂³⋊₃D₄.₅C₂, C₂².₄₁C₂²≀C₂, C₂³.₉D₄⋊₁₂C₂, C₂³.₃₄D₄⋊₈C₂, (C₂³×C₄).₂₆C₂², C₂³.₁₂₇(C₄○D₄), C₂².₁₂(C₄⋊₁D₄), C₂².₆₅(C₄⋊D₄), C₂.₂₇(C₂³⋊₂D₄), (C₂²×D₄).₇₁C₂², C₂.₁₆(C₂³.₇D₄), (C₂×C₂³⋊C₄)⋊₉C₂, (C₂×C₂².D₄)⋊₁C₂, (C₂×C₂²⋊C₄).₁₀₃C₂², SmallGroup(128,754)

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

Derived series

C₁ — C₂⁴ — C₂⁴.₃₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — C₂×C₂².D₄ — C₂⁴.₃₁D₄

Lower central

C₁ — C₂ — C₂⁴ — C₂⁴.₃₁D₄

Upper central

C₁ — C₂² — C₂⁴ — C₂⁴.₃₁D₄

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₃₁D₄

Generators and relations for C₂⁴.₃₁D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=d, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, bc=cb, ebe^-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de^-1 >

Subgroups: 528 in 215 conjugacy classes, 46 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₉D₄, C₂³.₃₄D₄, C₂×C₂³⋊C₄, C₂×C₂².D₄, C₂³⋊₃D₄, C₂⁴.₃₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₂³⋊₂D₄, C₂³.₇D₄, C₂⁴.₃₁D₄

Character table of C₂⁴.₃₁D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N
size	1	1	1	1	2	2	2	2	2	2	8	8	4	4	4	4	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	0	0	0	0	0	2	0	0	0	0	0	-2	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	-2	0	0	0	0	0	2	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	2	-2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	2	-2	2	-2	2	0	0	0	0	0	0	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	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	-2	-2	-2	-2	2	0	2	0	0	0	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	-2	2	-2	2	2	-2	-2	2	0	0	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	-2	-2	2	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	-2	2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	2	2	2	-2	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	2	0	orthogonal lifted from D₄
ρ₁₈	2	-2	-2	2	-2	-2	-2	2	2	2	0	0	0	0	0	0	0	0	0	0	-2	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	2	2	-2	-2	-2	-2	2	0	-2	0	0	0	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	-2	2	-2	-2	-2	2	2	2	0	0	0	0	0	0	0	0	0	0	2	0	-2	0	0	0	orthogonal lifted from 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	-2i	0	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	-2i	2i	-2i	2i	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	0	0	2i	-2i	2i	-2i	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	0	0	2i	-2i	-2i	2i	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	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂³.₇D₄

Smallest permutation representation of C₂⁴.₃₁D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

Matrix representation of C₂⁴.₃₁D₄ ►in GL₆(𝔽₅)

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

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

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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

C₂⁴.₃₁D₄ in GAP, Magma, Sage, TeX

C_2^4._{31}D_4

% in TeX

G:=Group("C2^4.31D4");

// GroupNames label

G:=SmallGroup(128,754);

// by ID

G=gap.SmallGroup(128,754);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,521,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴.₃₁D₄ in TeX

G = C24.31D4 order 128 = 27

31st non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₃₁D₄ order 128 = 2⁷

31^st non-split extension by C₂⁴ of D₄ acting faithfully