C2^4.33D4 - GroupNames

G = C₂⁴.₃₃D₄ order 128 = 2⁷

33^rd non-split extension by C₂⁴ of D₄ acting faithfully

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

Aliases: C₂⁴.₃₃D₄, C₂⁴.₁₈₁C₂³, C₂²⋊C₄⋊₇D₄, C₂.₁₈C₂≀C₂², (C₂²×C₄).₈₄D₄, C₂³.₃₄(C₂×D₄), C₂³⋊₃D₄.₆C₂, C₂³.₈Q₈⋊₂C₂, C₂³.₁₁(C₄○D₄), C₂³.₂₃D₄⋊₂C₂, C₂³.₉D₄⋊₁₅C₂, (C₂³×C₄).₂₈C₂², C₂².₂₂₆C₂²≀C₂, C₂².₆₈(C₄⋊D₄), (C₂²×D₄).₈₁C₂², C₂².₃₈(C₄.₄D₄), C₂.₁₈(C₂³.₇D₄), C₂.₁₇(C₂³.₁₀D₄), C₂².₃₄(C₂².D₄), (C₂×C₂³⋊C₄)⋊₁₀C₂, (C₂×C₂²⋊C₄).₂₀C₂², SmallGroup(128,776)

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

Derived series

C₁ — C₂⁴ — C₂⁴.₃₃D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₃₃D₄

Lower central

C₁ — C₂ — C₂⁴ — C₂⁴.₃₃D₄

Upper central

C₁ — C₂² — C₂⁴ — C₂⁴.₃₃D₄

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₃₃D₄

Generators and relations for C₂⁴.₃₃D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, faf=ab=ba, ac=ca, ad=da, eae^-1=abcd, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf=cd=dc, de=ed, df=fd, fef=be^-1 >

Subgroups: 504 in 191 conjugacy classes, 42 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×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, C₂³.₈Q₈, C₂³.₂₃D₄, C₂×C₂³⋊C₄, C₂³⋊₃D₄, C₂⁴.₃₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂³.₁₀D₄, C₂≀C₂², C₂³.₇D₄, C₂⁴.₃₃D₄

Character table of C₂⁴.₃₃D₄

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	0	0	2	0	-2	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-2	0	2	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	2	-2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2	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	0	-2	orthogonal lifted from D₄
ρ₁₃	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
ρ₁₄	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
ρ₁₅	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
ρ₁₆	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
ρ₁₇	2	-2	-2	2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	-2i	0	0	0	0	2i	0	complex lifted from C₄○D₄
ρ₁₈	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
ρ₁₉	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
ρ₂₀	2	-2	-2	2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	2i	0	0	0	0	-2i	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	-2	2	-2	-2	2	-2	2	2	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	-2	-2	2	-2	2	2	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂≀C₂²
ρ₂₄	4	4	-4	-4	0	0	0	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂≀C₂²
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 C₂⁴.₃₃D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

Matrix representation of C₂⁴.₃₃D₄ ►in GL₆(𝔽₅)

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

4	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
0	0	0	0	0	4

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

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

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

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

C₂⁴.₃₃D₄ in GAP, Magma, Sage, TeX

C_2^4._{33}D_4

% in TeX

G:=Group("C2^4.33D4");

// GroupNames label

G:=SmallGroup(128,776);

// by ID

G=gap.SmallGroup(128,776);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,1411,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴.₃₃D₄ in TeX

G = C24.33D4 order 128 = 27

33rd non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₃₃D₄ order 128 = 2⁷

33^rd non-split extension by C₂⁴ of D₄ acting faithfully