C2^4.96D4

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

Subgroups: 756 in 350 conjugacy classes, 108 normal (8 characteristic)
C₁, C₂, C₂^[×6], C₂^[×10], C₄^[×12], C₂², C₂²^[×14], C₂²^[×50], C₂×C₄^[×52], C₂³, C₂³^[×14], C₂³^[×50], C₂²⋊C₄^[×26], C₄⋊C₄^[×4], C₂²×C₄^[×12], C₂²×C₄^[×20], C₂⁴, C₂⁴^[×6], C₂⁴^[×10], C₂.C₄²^[×12], C₂×C₂²⋊C₄^[×14], C₂×C₂²⋊C₄^[×8], C₂×C₄⋊C₄^[×4], C₂³×C₄^[×4], C₂⁵, C₂⁴⋊₃C₄, C₂³.₃₄D₄^[×4], C₂³.₈Q₈^[×4], C₂³.₁₁D₄^[×4], C₂²×C₂²⋊C₄^[×2], C₂⁴.₉₆D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×8], C₂⁴, C₂².D₄^[×8], C₂²×D₄, C₂×C₄○D₄^[×4], 2₊⁽¹⁺⁴⁾^[×2], C₂×C₂².D₄^[×2], C₂³⋊₃D₄, C₂².₄₅C₂⁴^[×4], C₂⁴.₉₆D₄

Generators and relations
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 >

Smallest permutation representation
►On 32 points

Generators in S₃₂

(2 15)(4 13)(6 30)(8 32)(9 17)(10 23)(11 19)(12 21)(18 28)(20 26)(22 27)(24 25)

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

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

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

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

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