C2^4:7D4 - GroupNames

G = C₂⁴⋊₇D₄ order 128 = 2⁷

2^nd semidirect product of C₂⁴ and D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴⋊₇D₄, C₂⁵⋊₁C₂², C₂³.₃₀₃C₂⁴, C₂⁴.₅₆₀C₂³, C₂².₁₂₁2₊¹⁺⁴, C₂.₆D₄², (C₂×D₄)⋊₄₀D₄, C₂³⋊₅(C₂×D₄), (D₄×C₂³)⋊₂C₂, C₂²⋊₂C₂²≀C₂, C₂³⋊₂D₄⋊₃C₂, C₂⁴⋊₃C₄⋊₁₀C₂, (C₂³×C₄)⋊₁₈C₂², (C₂²×D₄)⋊₂C₂², C₂.₆(C₂³⋊₃D₄), C₂³.₂₃D₄⋊₂₀C₂, (C₂²×C₄).₇₈₅C₂³, C₂².₁₈₃(C₂²×D₄), C₂.C₄²⋊₁₄C₂², (C₂×C₄)⋊₈(C₂×D₄), (C₂×C₂²≀C₂)⋊₂C₂, C₂.₁₀(C₂×C₂²≀C₂), (C₂×C₂²⋊C₄)⋊₈C₂², SmallGroup(128,1135)

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

Derived series

C₁ — C₂³ — C₂⁴⋊₇D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — D₄×C₂³ — C₂⁴⋊₇D₄

Lower central

C₁ — C₂³ — C₂⁴⋊₇D₄

Upper central

C₁ — C₂³ — C₂⁴⋊₇D₄

Jennings

C₁ — C₂³ — C₂⁴⋊₇D₄

Generators and relations for C₂⁴⋊₇D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, faf=ac=ca, ad=da, eae^-1=acd, ebe^-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1780 in 782 conjugacy classes, 140 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂²≀C₂, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁵, C₂⁴⋊₃C₄, C₂³.₂₃D₄, C₂³⋊₂D₄, C₂×C₂²≀C₂, D₄×C₂³, C₂⁴⋊₇D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, 2₊¹⁺⁴, C₂×C₂²≀C₂, C₂³⋊₃D₄, D₄², C₂⁴⋊₇D₄

Smallest permutation representation of C₂⁴⋊₇D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	···	2O	2P	···	2Y	4A	···	4H	4I	4J	4K	4L
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	D₄	2₊¹⁺⁴
kernel	C₂⁴⋊₇D₄	C₂⁴⋊₃C₄	C₂³.₂₃D₄	C₂³⋊₂D₄	C₂×C₂²≀C₂	D₄×C₂³	C₂×D₄	C₂⁴	C₂²
# reps	1	1	4	4	4	2	16	4	2

Matrix representation of C₂⁴⋊₇D₄ ►in GL₆(𝔽₅)

0	1	0	0	0	0
1	0	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

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	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

4	0	0	0	0	0
0	4	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	1

0	1	0	0	0	0
4	0	0	0	0	0
0	0	3	2	0	0
0	0	0	2	0	0
0	0	0	0	0	4
0	0	0	0	1	0

0	4	0	0	0	0
4	0	0	0	0	0
0	0	2	3	0	0
0	0	4	3	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

C₂⁴⋊₇D₄ in GAP, Magma, Sage, TeX

C_2^4\rtimes_7D_4

% in TeX

G:=Group("C2^4:7D4");

// GroupNames label

G:=SmallGroup(128,1135);

// by ID

G=gap.SmallGroup(128,1135);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24⋊7D4 order 128 = 27

2nd semidirect product of C24 and D4 acting via D4/C2=C22

G = C₂⁴⋊₇D₄ order 128 = 2⁷

2^nd semidirect product of C₂⁴ and D₄ acting via D₄/C₂=C₂²