C2^4:13D4 - GroupNames

G = C₂⁴⋊₁₃D₄ order 128 = 2⁷

1^st semidirect product of C₂⁴ and D₄ acting via D₄/C₄=C₂

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

Aliases: C₂⁴⋊₁₃D₄, C₂⁵.₈₉C₂², C₂⁴.₁₁₈C₂³, C₂³.₇₄₇C₂⁴, C₄⋊₃C₂²≀C₂, (C₂⁴×C₄)⋊₇C₂, (C₂²×C₄)⋊₅₄D₄, C₂²⋊₃(C₄⋊D₄), C₂³.₆₂₉(C₂×D₄), (C₂²×D₄)⋊₁₆C₂², C₂³.₂₄₇(C₄○D₄), (C₂²×C₄).₂₅₇C₂³, (C₂³×C₄).₆₈₁C₂², C₂³.₇Q₈⋊₁₁₅C₂, C₂².₄₅₇(C₂²×D₄), C₂³.₂₃D₄⋊₁₁₁C₂, C₂.C₄²⋊₄₅C₂², C₂.₉₀(C₂².₁₉C₂⁴), (C₂×C₄⋊D₄)⋊₄₁C₂, (C₂×C₄⋊C₄)⋊₄₀C₂², C₂.₄₆(C₂×C₄⋊D₄), (C₂×C₂²≀C₂)⋊₁₈C₂, C₂.₃₀(C₂×C₂²≀C₂), (C₂×C₄).₁₂₀₂(C₂×D₄), (C₂×C₂²⋊C₄)⋊₃₄C₂², C₂².₅₈₈(C₂×C₄○D₄), SmallGroup(128,1579)

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

Derived series

C₁ — C₂³ — C₂⁴⋊₁₃D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴×C₄ — C₂⁴⋊₁₃D₄

Lower central

C₁ — C₂³ — C₂⁴⋊₁₃D₄

Upper central

C₁ — C₂³ — C₂⁴⋊₁₃D₄

Jennings

C₁ — C₂³ — C₂⁴⋊₁₃D₄

Generators and relations for C₂⁴⋊₁₃D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, faf=ac=ca, ad=da, ae=ea, bc=cb, fbf=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1188 in 612 conjugacy classes, 144 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²≀C₂, C₄⋊D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂³.₇Q₈, C₂³.₂₃D₄, C₂×C₂²≀C₂, C₂×C₄⋊D₄, C₂⁴×C₄, C₂⁴⋊₁₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²≀C₂, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₂²≀C₂, C₂×C₄⋊D₄, C₂².₁₉C₂⁴, C₂⁴⋊₁₃D₄

Smallest permutation representation of C₂⁴⋊₁₃D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	8	8	2	···	2	8	···	8

44 irreducible representations

dim	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	C₄○D₄
kernel	C₂⁴⋊₁₃D₄	C₂³.₇Q₈	C₂³.₂₃D₄	C₂×C₂²≀C₂	C₂×C₄⋊D₄	C₂⁴×C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	3	6	2	3	1	12	4	12

Matrix representation of C₂⁴⋊₁₃D₄ ►in GL₆(𝔽₅)

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	4	0
0	0	0	0	0	1

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	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	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

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

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

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

C₂⁴⋊₁₃D₄ in GAP, Magma, Sage, TeX

C_2^4\rtimes_{13}D_4

% in TeX

G:=Group("C2^4:13D4");

// GroupNames label

G:=SmallGroup(128,1579);

// by ID

G=gap.SmallGroup(128,1579);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,2019]);

// 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,a*e=e*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*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⋊13D4 order 128 = 27

1st semidirect product of C24 and D4 acting via D4/C4=C2

G = C₂⁴⋊₁₃D₄ order 128 = 2⁷

1^st semidirect product of C₂⁴ and D₄ acting via D₄/C₄=C₂