C2^4.110D4 - GroupNames

G = C₂⁴.₁₁₀D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₁₀D₄, (C₂×C₈)⋊₁₀D₄, C₄ ○(C₈⋊₂D₄), C₄ ○(C₈⋊D₄), C₈.₂₁(C₂×D₄), C₈⋊₂D₄⋊₃₅C₂, C₈⋊D₄⋊₅₇C₂, C₄ ○(C₈.D₄), C₈.D₄⋊₃₅C₂, (C₂×D₈)⋊₄₇C₂², C₄⋊C₄.₂₃C₂³, C₄.Q₈⋊₅₀C₂², C₂.D₈⋊₆₉C₂², (C₂×C₄).₂₅₈C₂⁴, (C₂×C₈).₂₅₀C₂³, (C₂×Q₁₆)⋊₅₂C₂², (C₂×D₄).₆₁C₂³, C₄.₁₅₂(C₂²×D₄), C₂³.₂₃₈(C₂×D₄), (C₂²×C₄).₄₂₈D₄, (C₂×Q₈).₄₉C₂³, C₄.₂₁₂(C₄⋊D₄), D₄⋊C₄⋊₉₂C₂², C₂².₁₉C₂⁴⋊₇C₂, Q₈⋊C₄⋊₉₆C₂², (C₂×SD₁₆)⋊₅₅C₂², (C₂²×M₄(2))⋊₃C₂, C₄⋊D₄.₁₄₉C₂², C₂³.₂₄D₄⋊₃₉C₂, C₂³.₂₅D₄⋊₂₆C₂, C₂².₃₅(C₄⋊D₄), (C₂³×C₄).₅₅₀C₂², (C₂²×C₈).₂₅₈C₂², C₂².₅₁₈(C₂²×D₄), C₂²⋊Q₈.₁₅₄C₂², C₂.₁₄(D₈⋊C₂²), (C₂²×C₄).₁₅₃₇C₂³, C₄²⋊C₂.₁₀₇C₂², (C₂×M₄(2)).₂₆₃C₂², (C₂×C₄○D₈)⋊₁₆C₂, (C₂×C₄)○(C₈⋊D₄), (C₂×C₄)○(C₈⋊₂D₄), C₄.₂₅(C₂×C₄○D₄), (C₂×C₄)○(C₈.D₄), (C₂×C₄).₄₇₄(C₂×D₄), C₂.₇₆(C₂×C₄⋊D₄), (C₂×C₄).₇₀₄(C₄○D₄), (C₂×C₄○D₄).₁₂₄C₂², SmallGroup(128,1786)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₁₀D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×M₄(2) — C₂⁴.₁₁₀D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₁₀D₄

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂⁴.₁₁₀D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₁₀D₄

Generators and relations for C₂⁴.₁₁₀D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=f²=d, ab=ba, ac=ca, eae^-1=ad=da, faf^-1=acd, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 484 in 250 conjugacy classes, 100 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₄○D₈, C₂³×C₄, C₂×C₄○D₄, C₂³.₂₄D₄, C₂³.₂₅D₄, C₈⋊D₄, C₈⋊₂D₄, C₈.D₄, C₂².₁₉C₂⁴, C₂²×M₄(2), C₂×C₄○D₈, C₂⁴.₁₁₀D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄⋊D₄, D₈⋊C₂², C₂⁴.₁₁₀D₄

Smallest permutation representation of C₂⁴.₁₁₀D₄
►On 32 points

Generators in S₃₂

(2 6)(4 8)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(18 22)(20 24)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₈⋊C₂²

kernel

C₂⁴.₁₁₀D₄

C₂³.₂₄D₄

C₂³.₂₅D₄

C₈⋊D₄

C₈⋊₂D₄

C₈.D₄

C₂².₁₉C₂⁴

C₂²×M₄(2)

C₂×C₄○D₈

C₂×C₈

C₂²×C₄

C₂⁴

C₂×C₄

C₂

# reps

Matrix representation of C₂⁴.₁₁₀D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	0	16

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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	1	0	0
0	0	4	0	0	0
0	0	0	0	0	4
0	0	0	0	16	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	0	4
0	0	0	0	16	0
0	0	0	1	0	0
0	0	4	0	0	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,4,0,0,0] >;

C₂⁴.₁₁₀D₄ in GAP, Magma, Sage, TeX

C_2^4._{110}D_4

% in TeX

G:=Group("C2^4.110D4");

// GroupNames label

G:=SmallGroup(128,1786);

// by ID

G=gap.SmallGroup(128,1786);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,248,2804,172]);

// 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,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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 = C24.110D4 order 128 = 27

65th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₁₀D₄ order 128 = 2⁷

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