C2^2xC4wrC2 - GroupNames

G = C₂²xC₄wrC₂ order 128 = 2⁷

Direct product of C₂² and C₄wrC₂

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

Aliases: C₂²xC₄wrC₂, C₄²:₁₉C₂³, C₂⁴.₁₇₁D₄, M₄(2):₁₁C₂³, D₄:₆(C₂²xC₄), Q₈:₆(C₂²xC₄), (C₂²xD₄):₂₈C₄, C₄.₁₀(C₂³xC₄), (C₂²xQ₈):₂₂C₄, (C₂xC₄).₁₈₀C₂⁴, (C₂²xC₄²):₁₈C₂, (C₂xC₄²):₈₆C₂², C₄oD₄.₁₉C₂³, C₂³.₆₄₀(C₂xD₄), (C₂²xC₄).₈₂₀D₄, C₄.₁₈₀(C₂²xD₄), C₂².₂₇(C₂²xD₄), (C₂xM₄(2)):₇₂C₂², (C₂²xM₄(2)):₂₁C₂, (C₂³xC₄).₆₉₁C₂², C₂³.₁₃₂(C₂²:C₄), (C₂²xC₄).₁₄₉₈C₂³, C₄ o(C₂xC₄wrC₂), (C₂xC₄)o C₄wrC₂, C₄oD₄:₁₅(C₂xC₄), (C₂xC₄oD₄):₂₁C₄, (C₂xD₄):₅₀(C₂xC₄), (C₂xQ₈):₄₁(C₂xC₄), C₄.₇₆(C₂xC₂²:C₄), (C₂xC₄).₁₅₆₄(C₂xD₄), (C₂²xC₄).₄₁₆(C₂xC₄), (C₂xC₄).₄₆₂(C₂²xC₄), (C₂²xC₄oD₄).₂₁C₂, C₂².₂₃(C₂xC₂²:C₄), C₂.₄₂(C₂²xC₂²:C₄), (C₂xC₄).₂₈₅(C₂²:C₄), (C₂xC₄oD₄).₂₇₅C₂², (C₂xC₄)o(C₂xC₄wrC₂), SmallGroup(128,1631)

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

Derived series

C₁ — C₄ — C₂²xC₄wrC₂

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄oD₄ — C₂²xC₄wrC₂

Lower central

C₁ — C₂ — C₄ — C₂²xC₄wrC₂

Upper central

C₁ — C₂²xC₄ — C₂³xC₄ — C₂²xC₄wrC₂

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — C₂²xC₄wrC₂

Generators and relations for C₂²xC₄wrC₂
G = < a,b,c,d,e | a²=b²=c⁴=d²=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c^-1, ce=ec, ede^-1=c^-1d >

Subgroups: 716 in 428 conjugacy classes, 180 normal (17 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₄², C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₄oD₄, C₄oD₄, C₂⁴, C₂⁴, C₄wrC₂, C₂xC₄², C₂xC₄², C₂²xC₈, C₂xM₄(2), C₂xM₄(2), C₂³xC₄, C₂³xC₄, C₂²xD₄, C₂²xD₄, C₂²xQ₈, C₂xC₄oD₄, C₂xC₄oD₄, C₂xC₄wrC₂, C₂²xC₄², C₂²xM₄(2), C₂²xC₄oD₄, C₂²xC₄wrC₂
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂⁴, C₄wrC₂, C₂xC₂²:C₄, C₂³xC₄, C₂²xD₄, C₂xC₄wrC₂, C₂²xC₂²:C₄, C₂²xC₄wrC₂

Smallest permutation representation of C₂²xC₄wrC₂
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	···	4H	4I	···	4AB	4AC	4AD	4AE	4AF	8A	···	8H
order	1	2	···	2	2	2	2	2	2	2	2	2	4	···	4	4	···	4	4	4	4	4	8	···	8
size	1	1	···	1	2	2	2	2	4	4	4	4	1	···	1	2	···	2	4	4	4	4	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2
type	+	+	+	+	+				+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	C₄wrC₂
kernel	C₂²xC₄wrC₂	C₂xC₄wrC₂	C₂²xC₄²	C₂²xM₄(2)	C₂²xC₄oD₄	C₂²xD₄	C₂²xQ₈	C₂xC₄oD₄	C₂²xC₄	C₂⁴	C₂²
# reps	1	12	1	1	1	2	2	12	7	1	16

Matrix representation of C₂²xC₄wrC₂ ►in GL₅(F₁₇)

16	0	0	0	0
0	16	0	0	0
0	0	16	0	0
0	0	0	16	0
0	0	0	0	16

16	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	16	0	0	0
0	0	16	0	0
0	0	0	13	0
0	0	0	0	4

1	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	0	0	0	4
0	0	0	13	0

16	0	0	0	0
0	1	0	0	0
0	0	16	0	0
0	0	0	16	0
0	0	0	0	4

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,13,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,4,0],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,4] >;

C₂²xC₄wrC₂ in GAP, Magma, Sage, TeX

C_2^2\times C_4\wr C_2

% in TeX

G:=Group("C2^2xC4wrC2");

// GroupNames label

G:=SmallGroup(128,1631);

// by ID

G=gap.SmallGroup(128,1631);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,1411,172,124]);

// Polycyclic

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

// generators/relations

G = C22xC4wrC2 order 128 = 27

Direct product of C22 and C4wrC2

G = C₂²xC₄wrC₂ order 128 = 2⁷

Direct product of C₂² and C₄wrC₂