C2^4.105D4 - GroupNames

G = C₂⁴.₁₀₅D₄ order 128 = 2⁷

60^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₁₀₅D₄, D₄⋊₈(C₂×D₄), Q₈⋊₈(C₂×D₄), C₄○D₄⋊₁₄D₄, Q₈⋊D₄⋊₂C₂, C₄⋊C₄.₈C₂³, (C₂×C₈).₉C₂³, D₄⋊D₄⋊₁₄C₂, C₂²⋊D₈⋊₁₂C₂, (C₂×D₈)⋊₁₅C₂², C₂²⋊C₈⋊₅C₂², C₄.₄₃(C₂²×D₄), C₄.₁₀₆C₂²≀C₂, C₄⋊D₄⋊₅₂C₂², C₂²⋊₅(C₈⋊C₂²), C₂⁴.₄C₄⋊₅C₂, (C₂×C₄).₂₂₅C₂⁴, (C₂×SD₁₆)⋊₄C₂², (C₂×D₄).₂₈C₂³, C₂³.₆₄₈(C₂×D₄), (C₂²×C₄).₇₈₇D₄, D₄⋊C₄⋊₁₀C₂², Q₈⋊C₄⋊₁₃C₂², C₂².₁₇C₂²≀C₂, C₂³.₃₆D₄⋊₁C₂, (C₂²×D₄)⋊₁₇C₂², (C₂×M₄(2))⋊₂C₂², (C₂×Q₈).₃₅₃C₂³, (C₂²×Q₈)⋊₅₄C₂², C₂.₈(D₈⋊C₂²), (C₂³×C₄).₅₄₅C₂², (C₂²×C₄).₉₆₃C₂³, C₂².₄₈₅(C₂²×D₄), (C₂×C₈⋊C₂²)⋊₇C₂, (C₂×C₄⋊D₄)⋊₄₅C₂, (C₂×C₄⋊C₄)⋊₄₆C₂², (C₂×C₄).₄₅₂(C₂×D₄), C₂.₁₁(C₂×C₈⋊C₂²), C₂.₄₃(C₂×C₂²≀C₂), (C₂²×C₄○D₄)⋊₁₀C₂, (C₂×C₄○D₄)⋊₆₄C₂², SmallGroup(128,1738)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₀₅D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂⁴.₁₀₅D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₀₅D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₀₅D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₀₅D₄

Generators and relations for C₂⁴.₁₀₅D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=f²=1, e⁴=d, ab=ba, eae^-1=faf=ac=ca, ad=da, bc=cb, ebe^-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de³ >

Subgroups: 844 in 403 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), D₈, SD₁₆, 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, C₄⋊D₄, C₄⋊D₄, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂⁴.₄C₄, C₂³.₃₆D₄, C₂²⋊D₈, Q₈⋊D₄, D₄⋊D₄, C₂×C₄⋊D₄, C₂×C₈⋊C₂², C₂²×C₄○D₄, C₂⁴.₁₀₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₈⋊C₂², C₂²×D₄, C₂×C₂²≀C₂, C₂×C₈⋊C₂², D₈⋊C₂², C₂⁴.₁₀₅D₄

Smallest permutation representation of C₂⁴.₁₀₅D₄
►On 32 points

Generators in S₃₂

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

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

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

(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)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₈⋊C₂²

D₈⋊C₂²

kernel

C₂⁴.₁₀₅D₄

C₂⁴.₄C₄

C₂³.₃₆D₄

C₂²⋊D₈

Q₈⋊D₄

D₄⋊D₄

C₂×C₄⋊D₄

C₂×C₈⋊C₂²

C₂²×C₄○D₄

C₂²×C₄

C₄○D₄

C₂⁴

C₂²

C₂

# reps

Matrix representation of C₂⁴.₁₀₅D₄ ►in GL₆(ℤ)

1	-1	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	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	-1	0
0	0	-1	1	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	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	-1	0
0	0	0	0	0	-1

1	0	0	0	0	0
2	-1	0	0	0	0
0	0	-1	1	1	-2
0	0	0	0	1	0
0	0	1	0	0	0
0	0	1	0	0	1

-1	0	0	0	0	0
-2	1	0	0	0	0
0	0	1	0	0	0
0	0	0	-1	0	0
0	0	-1	1	1	-2
0	0	-1	0	0	-1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,-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,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-1,0,0,0,1,0,1,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[1,2,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,-2,0,0,1],[-1,-2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,-2,-1] >;

C₂⁴.₁₀₅D₄ in GAP, Magma, Sage, TeX

C_2^4._{105}D_4

% in TeX

G:=Group("C2^4.105D4");

// GroupNames label

G:=SmallGroup(128,1738);

// by ID

G=gap.SmallGroup(128,1738);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24.105D4 order 128 = 27

60th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₀₅D₄ order 128 = 2⁷

60^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²