C2^4.106D4 - GroupNames

G = C₂⁴.₁₀₆D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₀₆D₄, C₄○D₄.₅₂D₄, D₄.₄₂(C₂×D₄), C₄⋊C₄.₉C₂³, Q₈.₄₂(C₂×D₄), C₂²⋊SD₁₆⋊₂C₂, C₂²⋊C₈⋊₆C₂², (C₂×C₈).₁₀C₂³, C₄.₄₄(C₂²×D₄), C₄.₁₀₇C₂²≀C₂, D₄.₇D₄⋊₁₄C₂, (C₂×C₄).₂₂₆C₂⁴, C₂⁴.₄C₄⋊₆C₂, C₂²⋊Q₁₆⋊₁₂C₂, (C₂×Q₁₆)⋊₁₄C₂², (C₂×SD₁₆)⋊₅C₂², C₂³.₆₄₉(C₂×D₄), (C₂²×C₄).₇₈₈D₄, C₂²⋊Q₈⋊₆₄C₂², (C₂×Q₈).₂₁C₂³, D₄⋊C₄⋊₁₁C₂², Q₈⋊C₄⋊₁₄C₂², C₂².₁₈C₂²≀C₂, (C₂×D₄).₃₈₁C₂³, C₂³.₃₆D₄⋊₂C₂, C₂²⋊₅(C₈.C₂²), (C₂×M₄(2))⋊₃C₂², (C₂²×Q₈)⋊₁₃C₂², C₂.₉(D₈⋊C₂²), (C₂³×C₄).₅₄₆C₂², (C₂²×C₄).₉₆₄C₂³, C₂².₄₈₆(C₂²×D₄), (C₂²×D₄).₅₆₅C₂², (C₂×C₄⋊C₄)⋊₄₇C₂², (C₂×C₄).₄₅₃(C₂×D₄), (C₂×C₈.C₂²)⋊₇C₂, (C₂×C₂²⋊Q₈)⋊₅₃C₂, C₂.₄₄(C₂×C₂²≀C₂), C₂.₁₁(C₂×C₈.C₂²), (C₂²×C₄○D₄).₂₄C₂, (C₂×C₄○D₄).₂₉₇C₂², SmallGroup(128,1739)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₀₆D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂⁴.₁₀₆D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₀₆D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₀₆D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₀₆D₄

Generators and relations for C₂⁴.₁₀₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=f²=d, ab=ba, eae^-1=faf^-1=ac=ca, ad=da, bc=cb, ebe^-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 716 in 377 conjugacy classes, 110 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂⁴.₄C₄, C₂³.₃₆D₄, C₂²⋊SD₁₆, C₂²⋊Q₁₆, D₄.₇D₄, C₂×C₂²⋊Q₈, C₂×C₈.C₂², C₂²×C₄○D₄, C₂⁴.₁₀₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₈.C₂², C₂²×D₄, C₂×C₂²≀C₂, C₂×C₈.C₂², D₈⋊C₂², C₂⁴.₁₀₆D₄

Smallest permutation representation of C₂⁴.₁₀₆D₄
►On 32 points

Generators in S₃₂

(2 30)(4 32)(6 26)(8 28)(9 20)(11 22)(13 24)(15 18)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	···	2L	4A	···	4F	4G	···	4K	4L	4M	4N	4O	8A	8B	8C	8D
order	1	2	2	2	2	2	2	2	2	···	2	4	···	4	4	···	4	4	4	4	4	8	8	8	8
size	1	1	1	1	2	2	2	2	4	···	4	2	···	2	4	···	4	8	8	8	8	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₈.C₂²

D₈⋊C₂²

kernel

C₂⁴.₁₀₆D₄

C₂⁴.₄C₄

C₂³.₃₆D₄

C₂²⋊SD₁₆

C₂²⋊Q₁₆

D₄.₇D₄

C₂×C₂²⋊Q₈

C₂×C₈.C₂²

C₂²×C₄○D₄

C₂²×C₄

C₄○D₄

C₂⁴

C₂²

C₂

# reps

Matrix representation of C₂⁴.₁₀₆D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
6	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	16

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

11	2	0	0	0	0
7	6	0	0	0	0
0	0	0	0	12	12
0	0	0	0	12	5
0	0	12	5	0	0
0	0	5	5	0	0

11	2	0	0	0	0
8	6	0	0	0	0
0	0	12	5	0	0
0	0	5	5	0	0
0	0	0	0	12	12
0	0	0	0	12	5

G:=sub<GL(6,GF(17))| [1,6,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[11,7,0,0,0,0,2,6,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,12,0,0,0,0,12,5,0,0],[11,8,0,0,0,0,2,6,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,12,12,0,0,0,0,12,5] >;

C₂⁴.₁₀₆D₄ in GAP, Magma, Sage, TeX

C_2^4._{106}D_4

% in TeX

G:=Group("C2^4.106D4");

// GroupNames label

G:=SmallGroup(128,1739);

// by ID

G=gap.SmallGroup(128,1739);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,248,2804,1411,172]);

// Polycyclic

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

// generators/relations

G = C24.106D4 order 128 = 27

61st non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₀₆D₄ order 128 = 2⁷

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