C4xC2^2wrC2 - GroupNames

G = C₄xC₂²wrC₂ order 128 = 2⁷

Direct product of C₄ and C₂²wrC₂

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

Aliases: C₄xC₂²wrC₂, C₂⁵.₈₇C₂², C₂³.₁₈₁C₂⁴, C₂⁴.₅₃₆C₂³, (C₂⁴xC₄):₂C₂, C₂²:₄(C₄xD₄), C₂⁴:₁₅(C₂xC₄), (C₂²xC₄):₅₂D₄, (C₂³xC₄):₃C₂², C₂³:₁(C₂²xC₄), C₄ o₂(C₂⁴:₃C₄), C₂⁴:₃C₄:₃₀C₂, (C₂xC₄²):₁₂C₂², C₂³.₆₀₀(C₂xD₄), C₂².₇₂(C₂³xC₄), C₄ o₃(C₂³.₈Q₈), C₂².₇₇(C₂²xD₄), C₂³.₂₁₆(C₄oD₄), C₄ o₃(C₂³.₂₃D₄), (C₂²xC₄).₇₄₆C₂³, C₂³.₈Q₈:₁₅₀C₂, C₂³.₂₃D₄:₁₁₆C₂, C₂.C₄²:₆₀C₂², C₂.₅(C₂².₁₉C₂⁴), (C₂²xD₄).₄₆₇C₂², (C₂xC₄xD₄):₂C₂, C₂.₁₃(C₂xC₄xD₄), (C₂xD₄):₂₈(C₂xC₄), (C₂xC₄):₁(C₂²xC₄), (C₄xC₂²:C₄):₂₇C₂, C₂²:C₄:₂₄(C₂xC₄), C₂.₄(C₂xC₂²wrC₂), (C₂xC₄:C₄):₁₀₁C₂², (C₂xC₄).₁₅₅₈(C₂xD₄), C₂².₇₃(C₂xC₄oD₄), (C₂xC₄)o₂(C₂⁴:₃C₄), (C₂xC₂²wrC₂).₁₈C₂, (C₂xC₂²:C₄):₇₁C₂², (C₂xC₄)o₂(C₂³.₈Q₈), (C₂xC₄)o₂(C₂³.₂₃D₄), (C₂²xC₄)o(C₂³.₈Q₈), (C₂²xC₄)o(C₂³.₂₃D₄), (C₂xC₄)o(C₂xC₂²wrC₂), SmallGroup(128,1031)

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

Derived series

C₁ — C₂² — C₄xC₂²wrC₂

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂³xC₄ — C₂⁴xC₄ — C₄xC₂²wrC₂

Lower central

C₁ — C₂² — C₄xC₂²wrC₂

Upper central

C₁ — C₂²xC₄ — C₄xC₂²wrC₂

Jennings

C₁ — C₂³ — C₄xC₂²wrC₂

Generators and relations for C₄xC₂²wrC₂
G = < a,b,c,d,e,f | a⁴=b²=c²=d²=e²=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1068 in 612 conjugacy classes, 184 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂xC₄², C₂xC₂²:C₄, C₂xC₄:C₄, C₄xD₄, C₂²wrC₂, C₂³xC₄, C₂³xC₄, C₂³xC₄, C₂²xD₄, C₂⁵, C₄xC₂²:C₄, C₂⁴:₃C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂xC₄xD₄, C₂xC₂²wrC₂, C₂⁴xC₄, C₄xC₂²wrC₂
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²xC₄, C₂xD₄, C₄oD₄, C₂⁴, C₄xD₄, C₂²wrC₂, C₂³xC₄, C₂²xD₄, C₂xC₄oD₄, C₂xC₄xD₄, C₂xC₂²wrC₂, C₂².₁₉C₂⁴, C₄xC₂²wrC₂

Smallest permutation representation of C₄xC₂²wrC₂
►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 21)(2 22)(3 23)(4 24)(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(25 30)(26 31)(27 32)(28 29)

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	4A	···	4H	4I	···	4T	4U	···	4AH
order	1	2	···	2	2	···	2	2	2	4	···	4	4	···	4	4	···	4
size	1	1	···	1	2	···	2	4	4	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	D₄	C₄oD₄
kernel	C₄xC₂²wrC₂	C₄xC₂²:C₄	C₂⁴:₃C₄	C₂³.₈Q₈	C₂³.₂₃D₄	C₂xC₄xD₄	C₂xC₂²wrC₂	C₂⁴xC₄	C₂²wrC₂	C₂²xC₄	C₂³
# reps	1	3	1	3	3	3	1	1	16	12	12

Matrix representation of C₄xC₂²wrC₂ ►in GL₅(F₅)

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

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

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

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

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

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

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

C₄xC₂²wrC₂ in GAP, Magma, Sage, TeX

C_4\times C_2^2\wr C_2

% in TeX

G:=Group("C4xC2^2wrC2");

// GroupNames label

G:=SmallGroup(128,1031);

// by ID

G=gap.SmallGroup(128,1031);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,248]);

// Polycyclic

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

// generators/relations

G = C4xC22wrC2 order 128 = 27

Direct product of C4 and C22wrC2

G = C₄xC₂²wrC₂ order 128 = 2⁷

Direct product of C₄ and C₂²wrC₂