C2^4.118D4 - GroupNames

G = C₂⁴.₁₁₈D₄ order 128 = 2⁷

73^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₁₁₈D₄, (C₂×C₈).₃₅C₂³, C₂.D₈⋊₂₁C₂², C₄.Q₈⋊₁₀C₂², C₄⋊C₄.₃₉₃C₂³, (C₂×C₄).₂₉₃C₂⁴, C₂³.₂₄₄(C₂×D₄), (C₂²×C₄).₄₄₄D₄, (C₂×Q₈).₆₉C₂³, Q₈⋊C₄⋊₂₂C₂², C₂³.₄₇D₄⋊₂C₂, C₂²⋊C₈.₁₆C₂², C₂⁴.₄C₄.₂C₂, M₄(2)⋊C₄⋊₂₅C₂, C₂³.₃₈D₄⋊₁₀C₂, C₂³.₄₈D₄⋊₁₂C₂, C₂³.₂₀D₄⋊₁₃C₂, (C₂³×C₄).₅₆₃C₂², C₂².₅₅₃(C₂²×D₄), C₂²⋊Q₈.₁₆₂C₂², C₂.₂₄(D₈⋊C₂²), (C₂²×C₄).₁₀₀₉C₂³, C₂².₃₅(C₈.C₂²), (C₂×M₄(2)).₇₅C₂², (C₂²×Q₈).₂₉₂C₂², C₄²⋊C₂.₃₁₈C₂², C₄.₁₀₀(C₂².D₄), C₂².₆₅(C₂².D₄), C₄.₁₀₃(C₂×C₄○D₄), (C₂×C₄).₄₈₈(C₂×D₄), C₂.₂₉(C₂×C₈.C₂²), (C₂×C₂²⋊Q₈).₅₇C₂, (C₂×C₄).₄₈₈(C₄○D₄), (C₂×C₄⋊C₄).₉₂₉C₂², (C₂×C₄²⋊C₂).₆₀C₂, C₂.₅₈(C₂×C₂².D₄), SmallGroup(128,1827)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₁₈D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₄²⋊C₂ — C₂×C₄²⋊C₂ — C₂⁴.₁₁₈D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₁₈D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₁₈D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₁₈D₄

Generators and relations for C₂⁴.₁₁₈D₄
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=ce³ >

Subgroups: 364 in 205 conjugacy classes, 94 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂²⋊C₈, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂²⋊Q₈, C₂²⋊Q₈, C₂×M₄(2), C₂³×C₄, C₂²×Q₈, C₂⁴.₄C₄, C₂³.₃₈D₄, M₄(2)⋊C₄, C₂³.₄₇D₄, C₂³.₄₈D₄, C₂³.₂₀D₄, C₂×C₄²⋊C₂, C₂×C₂²⋊Q₈, C₂⁴.₁₁₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₈.C₂², C₂²×D₄, C₂×C₄○D₄, C₂×C₂².D₄, C₂×C₈.C₂², D₈⋊C₂², C₂⁴.₁₁₈D₄

Smallest permutation representation of C₂⁴.₁₁₈D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	4A	···	4F	4G	···	4O	4P	4Q	4R	4S	8A	8B	8C	8D
order	1	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	2	···	2	4	···	4	8	8	8	8	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

C₈.C₂²

D₈⋊C₂²

kernel

C₂⁴.₁₁₈D₄

C₂⁴.₄C₄

C₂³.₃₈D₄

M₄(2)⋊C₄

C₂³.₄₇D₄

C₂³.₄₈D₄

C₂³.₂₀D₄

C₂×C₄²⋊C₂

C₂×C₂²⋊Q₈

C₂²×C₄

C₂⁴

C₂×C₄

C₂²

C₂

# reps

Matrix representation of C₂⁴.₁₁₈D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	8	0
0	0	0	1	0	2
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	16	0	0	0
0	0	0	1	0	0
0	0	0	0	16	0
0	0	0	0	0	1

16	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	0
0	0	0	0	0	16

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

0	4	0	0	0	0
13	0	0	0	0	0
0	0	0	2	0	10
0	0	9	0	11	0
0	0	0	8	0	8
0	0	8	0	15	0

0	16	0	0	0	0
16	0	0	0	0	0
0	0	9	0	10	0
0	0	0	15	0	6
0	0	2	0	8	0
0	0	0	2	0	2

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,8,0,16,0,0,0,0,2,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,9,0,8,0,0,2,0,8,0,0,0,0,11,0,15,0,0,10,0,8,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,9,0,2,0,0,0,0,15,0,2,0,0,10,0,8,0,0,0,0,6,0,2] >;

C₂⁴.₁₁₈D₄ in GAP, Magma, Sage, TeX

C_2^4._{118}D_4

% in TeX

G:=Group("C2^4.118D4");

// GroupNames label

G:=SmallGroup(128,1827);

// by ID

G=gap.SmallGroup(128,1827);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,100,2019,248,4037,1027,124]);

// 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=c*e^3>;

// generators/relations

G = C24.118D4 order 128 = 27

73rd non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₁₈D₄ order 128 = 2⁷

73^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²