C2^4.96D4 - GroupNames

G = C₂⁴.₉₆D₄ order 128 = 2⁷

51^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₉₆D₄, C₂⁵.₄₆C₂², C₂³.₃₈₃C₂⁴, C₂⁴.₅₇₅C₂³, C₂².₁₈₆2₊¹⁺⁴, C₂³.₆₁₂(C₂×D₄), C₂⁴⋊₃C₄.₁₀C₂, (C₂³×C₄).₉₅C₂², C₂³.₈Q₈⋊₆₂C₂, C₂³.₃₀₅(C₄○D₄), C₂³.₃₄D₄⋊₃₀C₂, C₂³.₁₁D₄⋊₂₅C₂, C₂.₁₀(C₂³⋊₃D₄), (C₂²×C₄).₅₂₀C₂³, C₂².₂₆₃(C₂²×D₄), C₂.C₄²⋊₂₅C₂², C₂.₂₆(C₂².₄₅C₂⁴), C₂².₆₂(C₂².D₄), (C₂×C₄⋊C₄)⋊₂₀C₂², C₂².₂₆₀(C₂×C₄○D₄), (C₂²×C₂²⋊C₄).₁₃C₂, C₂.₂₈(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₆₁C₂², SmallGroup(128,1215)

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

Derived series

C₁ — C₂³ — C₂⁴.₉₆D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₂²⋊C₄ — C₂⁴.₉₆D₄

Lower central

C₁ — C₂³ — C₂⁴.₉₆D₄

Upper central

C₁ — C₂³ — C₂⁴.₉₆D₄

Jennings

C₁ — C₂³ — C₂⁴.₉₆D₄

Generators and relations for C₂⁴.₉₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=d, ab=ba, faf^-1=ac=ca, ad=da, eae^-1=acd, ebe^-1=fbf^-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=cde^-1 >

Subgroups: 756 in 350 conjugacy classes, 108 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂⁵, C₂⁴⋊₃C₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₁₁D₄, C₂²×C₂²⋊C₄, C₂⁴.₉₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂².D₄, C₂³⋊₃D₄, C₂².₄₅C₂⁴, C₂⁴.₉₆D₄

Smallest permutation representation of C₂⁴.₉₆D₄
►On 32 points

Generators in S₃₂

(2 18)(4 20)(5 15)(6 22)(7 13)(8 24)(9 16)(10 23)(11 14)(12 21)(25 32)(27 30)

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄	2₊¹⁺⁴
kernel	C₂⁴.₉₆D₄	C₂⁴⋊₃C₄	C₂³.₃₄D₄	C₂³.₈Q₈	C₂³.₁₁D₄	C₂²×C₂²⋊C₄	C₂⁴	C₂³	C₂²
# reps	1	1	4	4	4	2	4	16	2

Matrix representation of C₂⁴.₉₆D₄ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	4	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

4	0	0	0	0	0
0	4	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	4

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

4	0	0	0	0	0
0	4	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

0	3	0	0	0	0
2	0	0	0	0	0
0	0	0	3	0	0
0	0	3	0	0	0
0	0	0	0	0	2
0	0	0	0	3	0

2	0	0	0	0	0
0	3	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂⁴.₉₆D₄ in GAP, Magma, Sage, TeX

C_2^4._{96}D_4

% in TeX

G:=Group("C2^4.96D4");

// GroupNames label

G:=SmallGroup(128,1215);

// by ID

G=gap.SmallGroup(128,1215);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,100,675]);

// Polycyclic

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

// generators/relations

G = C24.96D4 order 128 = 27

51st non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₉₆D₄ order 128 = 2⁷

51^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²