C4xC2^3:C4 - GroupNames

G = C₄×C₂³⋊C₄ order 128 = 2⁷

Direct product of C₄ and C₂³⋊C₄

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄×C₂³⋊C₄, C₂³⋊₁C₄², C₂⁴.₁₆₃C₂³, (C₂³×C₄)⋊₃C₄, (C₂×C₄)⋊₁C₄², (C₂×C₄²)⋊₆C₄, C₂⁴.₂₅(C₂×C₄), C₂².₂₄(C₄×D₄), C₂³.₂(C₄○D₄), C₂³.₅₄₀(C₂×D₄), (C₂²×C₄).₆₆₉D₄, C₂².₉(C₂×C₄²), C₂³.₁(C₂²×C₄), C₄ ○₂(C₂³.₉D₄), C₂³.₉D₄⋊₁₆C₂, (C₂³×C₄).₁₈C₂², (C₂²×D₄).₄₄₄C₂², C₂².₂₀(C₄²⋊C₂), C₂.₅(C₂³.C₂³), (C₂×C₄⋊C₄)⋊₂₅C₄, (C₂×C₄×D₄).₉C₂, (C₄×C₂²⋊C₄)⋊₁C₂, C₂.₅(C₂×C₂³⋊C₄), C₂²⋊C₄⋊₂₃(C₂×C₄), (C₂×C₂²⋊C₄)⋊₁₄C₄, C₂.₁₂(C₄×C₂²⋊C₄), (C₂×D₄).₁₅₆(C₂×C₄), (C₂×C₂³⋊C₄).₁₁C₂, (C₂²×C₄).₄₃₅(C₂×C₄), (C₂×C₄)○(C₂³.₉D₄), (C₂×C₄).₃₉₆(C₂²⋊C₄), C₂².₁₁₄(C₂×C₂²⋊C₄), (C₂×C₂²⋊C₄).₄₀₈C₂², (C₂×C₄)○(C₂×C₂³⋊C₄), SmallGroup(128,486)

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

Derived series

C₁ — C₂² — C₄×C₂³⋊C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₄×C₂³⋊C₄

Lower central

C₁ — C₂ — C₂² — C₄×C₂³⋊C₄

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₄×C₂³⋊C₄

Jennings

C₁ — C₂ — C₂⁴ — C₄×C₂³⋊C₄

Generators and relations for C₄×C₂³⋊C₄
G = < a,b,c,d,e | a⁴=b²=c²=d²=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 436 in 206 conjugacy classes, 78 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂³⋊C₄, C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂³×C₄, C₂²×D₄, C₂³.₉D₄, C₄×C₂²⋊C₄, C₂×C₂³⋊C₄, C₂×C₄×D₄, C₄×C₂³⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₄², C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂³⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×C₂²⋊C₄, C₂×C₂³⋊C₄, C₂³.C₂³, C₄×C₂³⋊C₄

Smallest permutation representation of C₄×C₂³⋊C₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	4A	4B	4C	4D	4E	···	4J	4K	···	4AF
order	1	2	2	2	2	···	2	2	2	4	4	4	4	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	1	1	1	1	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+						+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄	D₄	C₄○D₄	C₂³⋊C₄	C₂³.C₂³
kernel	C₄×C₂³⋊C₄	C₂³.₉D₄	C₄×C₂²⋊C₄	C₂×C₂³⋊C₄	C₂×C₄×D₄	C₂³⋊C₄	C₂×C₄²	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₂³×C₄	C₂²×C₄	C₂³	C₄	C₂
# reps	1	2	2	2	1	16	2	2	2	2	4	4	2	2

Matrix representation of C₄×C₂³⋊C₄ ►in GL₆(𝔽₅)

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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	4
0	0	0	0	4	0

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	4

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

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

C₄×C₂³⋊C₄ in GAP, Magma, Sage, TeX

C_4\times C_2^3\rtimes C_4

% in TeX

G:=Group("C4xC2^3:C4");

// GroupNames label

G:=SmallGroup(128,486);

// by ID

G=gap.SmallGroup(128,486);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,4037]);

// Polycyclic

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

// generators/relations

G = C4×C23⋊C4 order 128 = 27

Direct product of C4 and C23⋊C4

G = C₄×C₂³⋊C₄ order 128 = 2⁷

Direct product of C₄ and C₂³⋊C₄