C2^4.95D4 - GroupNames

G = C₂⁴.₉₅D₄ order 128 = 2⁷

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

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂⁴.₉₅D₄, C₂⁵.₃₇C₂², C₂³.₃₁₂C₂⁴, C₂⁴.₅₆₂C₂³, C₂².₁₂₈2₊¹⁺⁴, C₂²⋊C₄⋊₃₃D₄, C₂³.₁₅₂(C₂×D₄), C₂.₁₈(D₄⋊₅D₄), C₂³.₁₀D₄⋊₉C₂, C₂³.₁₁D₄⋊₃C₂, C₂.₉(C₂³⋊₃D₄), (C₂²×C₄).₄₈C₂³, C₂³.₇Q₈⋊₃₃C₂, C₂³.₈Q₈⋊₂₈C₂, C₂³.₂₉₉(C₄○D₄), C₂³.₃₄D₄⋊₁₉C₂, C₂³.₂₃D₄⋊₂₇C₂, C₂².₇₃(C₄⋊D₄), (C₂³×C₄).₃₃₀C₂², C₂².₁₉₂(C₂²×D₄), C₂.C₄²⋊₂₀C₂², (C₂²×D₄).₁₁₈C₂², C₂²⋊₁(C₂².D₄), C₂.₁₁(C₂².₄₅C₂⁴), (C₂×C₄⋊C₄)⋊₁₃C₂², (C₂×C₄).₃₀₆(C₂×D₄), C₂.₁₆(C₂×C₄⋊D₄), (C₂×C₂²≀C₂).₈C₂, (C₂×C₂²⋊C₄)⋊₁₄C₂², (C₂²×C₂²⋊C₄)⋊₁₉C₂, C₂².₁₉₁(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₅C₂, C₂.₁₄(C₂×C₂².D₄), SmallGroup(128,1144)

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

Derived series

C₁ — C₂³ — C₂⁴.₉₅D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₂²⋊C₄ — C₂⁴.₉₅D₄

Lower central

C₁ — C₂³ — C₂⁴.₉₅D₄

Upper central

C₁ — C₂³ — C₂⁴.₉₅D₄

Jennings

C₁ — C₂³ — C₂⁴.₉₅D₄

Generators and relations for C₂⁴.₉₅D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=c, ab=ba, eae^-1=ac=ca, ad=da, faf^-1=acd, ebe^-1=fbf^-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 900 in 407 conjugacy classes, 116 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂⁵, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₂₃D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂²×C₂²⋊C₄, C₂×C₂²≀C₂, C₂×C₂².D₄, C₂⁴.₉₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂³⋊₃D₄, D₄⋊₅D₄, C₂².₄₅C₂⁴, C₂⁴.₉₅D₄

Smallest permutation representation of C₂⁴.₉₅D₄
►On 32 points

Generators in S₃₂

(2 24)(4 22)(5 20)(6 30)(7 18)(8 32)(9 29)(10 17)(11 31)(12 19)(14 26)(16 28)

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	···	2O	2P	2Q	2R	4A	···	4P	4Q	4R	4S
order	1	2	···	2	2	···	2	2	2	2	4	···	4	4	4	4
size	1	1	···	1	2	···	2	4	4	8	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₉₅D₄

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂²×C₂²⋊C₄

C₂×C₂²≀C₂

C₂×C₂².D₄

C₂²⋊C₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₉₅D₄ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	4	0
0	0	0	0	0	4

0	1	0	0	0	0
4	0	0	0	0	0
0	0	0	1	0	0
0	0	4	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	4	0	0	0	0
4	0	0	0	0	0
0	0	0	2	0	0
0	0	2	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂⁴.₉₅D₄ in GAP, Magma, Sage, TeX

C_2^4._{95}D_4

% in TeX

G:=Group("C2^4.95D4");

// GroupNames label

G:=SmallGroup(128,1144);

// by ID

G=gap.SmallGroup(128,1144);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,100,675]);

// Polycyclic

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

// generators/relations

G = C24.95D4 order 128 = 27

50th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₉₅D₄ order 128 = 2⁷

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