C2^4:8Q8 - GroupNames

G = C₂⁴⋊₈Q₈ order 128 = 2⁷

1^st semidirect product of C₂⁴ and Q₈ acting via Q₈/C₄=C₂

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

Aliases: C₂⁴⋊₈Q₈, C₂⁵.₉₀C₂², C₂³.₇₄₈C₂⁴, C₂⁴.₅₉₅C₂³, (C₂⁴×C₄).₁₁C₂, C₄.₁₀₅C₂²≀C₂, (C₂²×C₄).₇₇₀D₄, C₂³.₆₃₀(C₂×D₄), C₂²⋊₄(C₂²⋊Q₈), C₂³.₁₀₃(C₂×Q₈), C₂⁴⋊₃C₄.₁₇C₂, (C₂²×Q₈)⋊₁₁C₂², C₂³.₂₄₈(C₄○D₄), (C₂²×C₄).₂₅₈C₂³, (C₂³×C₄).₆₈₂C₂², C₂³.₈Q₈⋊₁₄₅C₂, C₂³.₇Q₈⋊₁₁₆C₂, C₂².₄₅₈(C₂²×D₄), C₂².₁₇₈(C₂²×Q₈), C₂.C₄²⋊₄₆C₂², C₂.₉₁(C₂².₁₉C₂⁴), (C₂×C₄⋊C₄)⋊₄₁C₂², (C₂×C₂²⋊Q₈)⋊₄₉C₂, C₂.₄₅(C₂×C₂²⋊Q₈), C₂.₃₁(C₂×C₂²≀C₂), (C₂×C₄).₁₂₀₃(C₂×D₄), C₂².₅₈₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₅₉C₂², SmallGroup(128,1580)

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

Derived series

C₁ — C₂³ — C₂⁴⋊₈Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴×C₄ — C₂⁴⋊₈Q₈

Lower central

C₁ — C₂³ — C₂⁴⋊₈Q₈

Upper central

C₁ — C₂³ — C₂⁴⋊₈Q₈

Jennings

C₁ — C₂³ — C₂⁴⋊₈Q₈

Generators and relations for C₂⁴⋊₈Q₈
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=e², ab=ba, faf^-1=ac=ca, ad=da, ae=ea, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 948 in 534 conjugacy classes, 144 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂⁵, C₂⁴⋊₃C₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂×C₂²⋊Q₈, C₂⁴×C₄, C₂⁴⋊₈Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²≀C₂, C₂²⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₂²≀C₂, C₂×C₂²⋊Q₈, C₂².₁₉C₂⁴, C₂⁴⋊₈Q₈

Smallest permutation representation of C₂⁴⋊₈Q₈
►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 27)(22 28)(23 25)(24 26)

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4P	4Q	···	4X
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	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	Q₈	C₄○D₄
kernel	C₂⁴⋊₈Q₈	C₂⁴⋊₃C₄	C₂³.₇Q₈	C₂³.₈Q₈	C₂×C₂²⋊Q₈	C₂⁴×C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	2	3	6	3	1	12	4	12

Matrix representation of C₂⁴⋊₈Q₈ ►in GL₆(𝔽₅)

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	4

1	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	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	4	0
0	0	0	0	0	4

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

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

C₂⁴⋊₈Q₈ in GAP, Magma, Sage, TeX

C_2^4\rtimes_8Q_8

% in TeX

G:=Group("C2^4:8Q8");

// GroupNames label

G:=SmallGroup(128,1580);

// by ID

G=gap.SmallGroup(128,1580);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,2019]);

// Polycyclic

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

// generators/relations

G = C24⋊8Q8 order 128 = 27

1st semidirect product of C24 and Q8 acting via Q8/C4=C2

G = C₂⁴⋊₈Q₈ order 128 = 2⁷

1^st semidirect product of C₂⁴ and Q₈ acting via Q₈/C₄=C₂