C2^4.166D4 - GroupNames

G = C₂⁴.₁₆₆D₄ order 128 = 2⁷

21^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂²=C₂

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₁₆₆D₄, C₂⁵.₉₁C₂², C₂³.₇₄₉C₂⁴, C₂⁴.₅₉₆C₂³, (C₂⁴×C₄)⋊₃C₂, (C₂²×C₄)⋊₅₃D₄, C₂⁴⋊₃C₄⋊₂₉C₂, C₂³.₆₃₁(C₂×D₄), C₂².₅₆C₂²≀C₂, C₂³.₂₄₉(C₄○D₄), C₂³.₃₄D₄⋊₆₃C₂, (C₂²×C₄).₂₅₉C₂³, (C₂³×C₄).₆₄₉C₂², C₂³.₈Q₈⋊₁₄₆C₂, C₂².₄₅₉(C₂²×D₄), C₂³.₂₃D₄⋊₁₁₂C₂, C₂.C₄²⋊₄₇C₂², (C₂²×D₄).₃₀₈C₂², C₂²⋊₄(C₂².D₄), C₂.₉₂(C₂².₁₉C₂⁴), (C₂×C₄⋊C₄)⋊₄₂C₂², C₂.₃₂(C₂×C₂²≀C₂), (C₂×C₄).₁₂₀₄(C₂×D₄), (C₂×C₂²≀C₂).₁₆C₂, (C₂×C₂²⋊C₄)⋊₃₅C₂², C₂².₅₉₀(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄₅C₂, C₂.₄₄(C₂×C₂².D₄), SmallGroup(128,1581)

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

Derived series

C₁ — C₂³ — C₂⁴.₁₆₆D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂⁴×C₄ — C₂⁴.₁₆₆D₄

Lower central

C₁ — C₂³ — C₂⁴.₁₆₆D₄

Upper central

C₁ — C₂³ — C₂⁴.₁₆₆D₄

Jennings

C₁ — C₂³ — C₂⁴.₁₆₆D₄

Generators and relations for C₂⁴.₁₆₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, eae^-1=faf=ac=ca, ad=da, bc=cb, ebe^-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce^-1 >

Subgroups: 1028 in 551 conjugacy classes, 132 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂×C₂²≀C₂, C₂×C₂².D₄, C₂⁴×C₄, C₂⁴.₁₆₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₂²≀C₂, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂⁴.₁₆₆D₄

Smallest permutation representation of C₂⁴.₁₆₆D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	4A	···	4P	4Q	···	4W
order	1	2	···	2	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	8	2	···	2	8	···	8

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	C₄○D₄
kernel	C₂⁴.₁₆₆D₄	C₂⁴⋊₃C₄	C₂³.₃₄D₄	C₂³.₈Q₈	C₂³.₂₃D₄	C₂×C₂²≀C₂	C₂×C₂².D₄	C₂⁴×C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	1	2	4	4	1	2	1	8	4	16

Matrix representation of C₂⁴.₁₆₆D₄ ►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	4

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	0	0	0	1	0
0	0	0	0	0	4

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	4	0
0	0	0	0	0	4

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

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

0	1	0	0	0	0
1	0	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

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

C₂⁴.₁₆₆D₄ in GAP, Magma, Sage, TeX

C_2^4._{166}D_4

% in TeX

G:=Group("C2^4.166D4");

// GroupNames label

G:=SmallGroup(128,1581);

// by ID

G=gap.SmallGroup(128,1581);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,100,2019]);

// Polycyclic

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

// generators/relations

G = C24.166D4 order 128 = 27

21st non-split extension by C24 of D4 acting via D4/C22=C2

G = C₂⁴.₁₆₆D₄ order 128 = 2⁷

21^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂²=C₂