C2^4.23D4 - GroupNames

G = C₂⁴.₂₃D₄ order 128 = 2⁷

23^rd non-split extension by C₂⁴ of D₄ acting faithfully

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

Aliases: C₂⁴.₂₃D₄, C₄.₈₆(C₄×D₄), C₄²⋊₁₀(C₂×C₄), C₄⋊C₄.₃₁₆D₄, (C₂×D₄).₇₃D₄, C₄.₄D₄⋊₁₂C₄, C₄.₅(C₄⋊D₄), C₄²⋊₆C₄⋊₂₂C₂, C₂³.₅₆₀(C₂×D₄), C₂.₅(D₄.₉D₄), C₂².₉₉C₂²≀C₂, C₂³.₉(C₂²⋊C₄), C₂³.₃₈D₄⋊₂₂C₂, (C₂²×C₄).₆₈₃C₂³, (C₂×C₄²).₂₈₄C₂², C₂².₁₁C₂⁴.₅C₂, (C₂²×D₄).₁₉C₂², (C₂²×Q₈).₁₅C₂², C₄²⋊C₂.₂₀C₂², C₂.₂₄(C₂³.₂₃D₄), (C₂×M₄(2)).₁₈₀C₂², C₂².₄₈(C₂².D₄), (C₂×Q₈)⋊₈(C₂×C₄), (C₂×D₄).₇₆(C₂×C₄), (C₂×C₄).₅₅(C₄○D₄), (C₂×C₄).₁₀₀₄(C₂×D₄), (C₂×C₄.D₄).₈C₂, (C₂×C₄.₄D₄).₅C₂, (C₂×C₄).₁₈₅(C₂²×C₄), C₂².₄₀(C₂×C₂²⋊C₄), SmallGroup(128,617)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₂₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂².₁₁C₂⁴ — C₂⁴.₂₃D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₂₃D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₂₃D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂⁴.₂₃D₄

Generators and relations for C₂⁴.₂₃D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=f²=d, ab=ba, ac=ca, ad=da, eae^-1=faf^-1=acd, bc=cb, bd=db, be=eb, bf=fb, ece^-1=cd=dc, cf=fc, de=ed, df=fd, fef^-1=be³ >

Subgroups: 404 in 180 conjugacy classes, 56 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₄.D₄, Q₈⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₄.₄D₄, C₄.₄D₄, C₂×M₄(2), C₂²×D₄, C₂²×Q₈, C₄²⋊₆C₄, C₂×C₄.D₄, C₂³.₃₈D₄, C₂².₁₁C₂⁴, C₂×C₄.₄D₄, C₂⁴.₂₃D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂³.₂₃D₄, D₄.₉D₄, C₂⁴.₂₃D₄

Smallest permutation representation of C₂⁴.₂₃D₄
►On 32 points

Generators in S₃₂

(2 28)(3 7)(4 26)(6 32)(8 30)(9 18)(11 24)(12 16)(13 22)(15 20)(17 21)(25 29)

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

C₄○D₄

D₄.₉D₄

kernel

C₂⁴.₂₃D₄

C₄²⋊₆C₄

C₂×C₄.D₄

C₂³.₃₈D₄

C₂².₁₁C₂⁴

C₂×C₄.₄D₄

C₄.₄D₄

C₄⋊C₄

C₂×D₄

C₂⁴

C₂×C₄

C₂

# reps

Matrix representation of C₂⁴.₂₃D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	13
0	0	0	16	0	0
0	0	0	0	1	1
0	0	0	0	0	16

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	4
0	0	0	16	0	13
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

14	5	0	0	0	0
12	3	0	0	0	0
0	0	0	16	0	13
0	0	0	1	4	4
0	0	4	13	0	1
0	0	0	8	0	16

3	5	0	0	0	0
12	14	0	0	0	0
0	0	4	0	16	0
0	0	0	13	1	1
0	0	0	0	13	0
0	0	0	0	8	4

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,13,0,1,16],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,4,13,0,1],[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],[14,12,0,0,0,0,5,3,0,0,0,0,0,0,0,0,4,0,0,0,16,1,13,8,0,0,0,4,0,0,0,0,13,4,1,16],[3,12,0,0,0,0,5,14,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,16,1,13,8,0,0,0,1,0,4] >;

C₂⁴.₂₃D₄ in GAP, Magma, Sage, TeX

C_2^4._{23}D_4

% in TeX

G:=Group("C2^4.23D4");

// GroupNames label

G:=SmallGroup(128,617);

// by ID

G=gap.SmallGroup(128,617);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,2804,1411,2028,1027,124]);

// 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,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;

// generators/relations

G = C24.23D4 order 128 = 27

23rd non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₂₃D₄ order 128 = 2⁷

23^rd non-split extension by C₂⁴ of D₄ acting faithfully