C2^4.16D4 - GroupNames

G = C₂⁴.₁₆D₄ order 128 = 2⁷

16^th non-split extension by C₂⁴ of D₄ acting faithfully

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

Aliases: C₂⁴.₁₆D₄, C₂³.₈SD₁₆, C₄⋊C₄.₁₆D₄, C₂³⋊C₈⋊₁₁C₂, (C₂×D₄).₁₉D₄, (C₂²×C₄).₅₂D₄, C₂.₉(D₄⋊₄D₄), C₂²⋊C₈⋊₃₉C₂², C₂³⋊₃D₄.₃C₂, C₂³.₅₃₀(C₂×D₄), C₂³.₄Q₈⋊₁C₂, C₂.₈(C₂²⋊SD₁₆), C₂².SD₁₆⋊₁₅C₂, C₄⋊D₄.₁₃C₂², (C₂²×C₄).₁₉C₂³, C₂².₃₃(C₂×SD₁₆), C₂².₁₄₀C₂²≀C₂, C₂³.₄₆D₄⋊₂₅C₂, C₂².₄₄(C₈⋊C₂²), C₂.C₄²⋊₆C₂², C₂.₁₀(C₂³.₇D₄), (C₂×C₄⋊C₄)⋊₂C₂², (C₂×C₄).₂₀₈(C₂×D₄), (C₂×C₂²⋊C₄).₉₉C₂², SmallGroup(128,345)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₁₆D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³⋊₃D₄ — C₂⁴.₁₆D₄

Lower central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₆D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₆D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₁₆D₄

Generators and relations for C₂⁴.₁₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=c, f²=dc=cd, eae^-1=faf^-1=ab=ba, ac=ca, ad=da, bc=cb, ebe^-1=bd=db, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de³ >

Subgroups: 420 in 144 conjugacy classes, 34 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂²⋊C₈, D₄⋊C₄, C₄.Q₈, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₂³⋊C₈, C₂².SD₁₆, C₂³.₄Q₈, C₂³.₄₆D₄, C₂³⋊₃D₄, C₂⁴.₁₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₂²≀C₂, C₂×SD₁₆, C₈⋊C₂², C₂²⋊SD₁₆, D₄⋊₄D₄, C₂³.₇D₄, C₂⁴.₁₆D₄

Character table of C₂⁴.₁₆D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D
size	1	1	1	1	2	2	4	4	8	8	4	4	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	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	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	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	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	0	0	2	2	-2	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	0	0	-2	0	-2	2	0	0	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	2	-2	-2	0	0	-2	-2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	0	0	-2	2	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	0	0	2	0	-2	2	0	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	2	2	2	2	0	0	-2	-2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	√-2	√-2	-√-2	-√-2	complex lifted from SD₁₆
ρ₁₆	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₉	4	-4	-4	4	0	0	0	0	0	0	0	0	-2	0	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₀	4	-4	-4	4	0	0	0	0	0	0	0	0	2	0	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
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 C₈⋊C₂²
ρ₂₂	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	0	0	0	complex lifted from C₂³.₇D₄

Smallest permutation representation of C₂⁴.₁₆D₄
►On 32 points

Generators in S₃₂

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

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

(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 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(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)

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

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

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

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

Matrix representation of C₂⁴.₁₆D₄ ►in GL₆(𝔽₁₇)

1	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	16	0
0	0	0	0	0	16

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

12	5	0	0	0	0
12	12	0	0	0	0
0	0	2	2	2	15
0	0	15	15	2	15
0	0	2	15	2	2
0	0	2	15	15	15

5	12	0	0	0	0
12	12	0	0	0	0
0	0	2	2	2	15
0	0	2	2	15	2
0	0	2	15	2	2
0	0	15	2	2	2

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[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],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,2,15,2,2,0,0,2,15,15,15,0,0,2,2,2,15,0,0,15,15,2,15],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,2,2,2,15,0,0,2,2,15,2,0,0,2,15,2,2,0,0,15,2,2,2] >;

C₂⁴.₁₆D₄ in GAP, Magma, Sage, TeX

C_2^4._{16}D_4

% in TeX

G:=Group("C2^4.16D4");

// GroupNames label

G:=SmallGroup(128,345);

// by ID

G=gap.SmallGroup(128,345);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,422,1123,570,521,136,1411]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=c,f^2=d*c=c*d,e*a*e^-1=f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^3>;

// generators/relations

Export

Character table of C₂⁴.₁₆D₄ in TeX

G = C24.16D4 order 128 = 27

16th non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₁₆D₄ order 128 = 2⁷

16^th non-split extension by C₂⁴ of D₄ acting faithfully