C4.C2^2wrC2 - GroupNames

G = C₄.C₂²≀C₂ order 128 = 2⁷

2^nd non-split extension by C₄ of C₂²≀C₂ acting via C₂²≀C₂/C₂×D₄=C₂

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

Aliases: C₄.₅₀C₂²≀C₂, (C₂³×C₄).₁₉C₄, (C₂×Q₈).₂₀₄D₄, (Q₈×C₂³).₂C₂, C₂⁴.₁₁₄(C₂×C₄), (C₂²×C₄).₂₆₀D₄, (C₂²×Q₈).₂₃C₄, C₂.₉(C₂⁴⋊₃C₄), C₂²⋊₂(C₄.₁₀D₄), C₂⁴.₄C₄.₁₅C₂, (C₂³×C₄).₂₂₉C₂², (C₂²×C₄).₆₅₄C₂³, C₂³.₁₈₃(C₂²×C₄), C₂³.₁₉₃(C₂²⋊C₄), (C₂²×Q₈).₃₇₈C₂², (C₂×M₄(2)).₁₄₇C₂², (C₂×C₄).₁₃₀₉(C₂×D₄), (C₂×C₄.₁₀D₄)⋊₁₃C₂, (C₂×C₄).₄₂(C₂²⋊C₄), (C₂²×C₄).₂₅₉(C₂×C₄), C₂.₂₂(C₂×C₄.₁₀D₄), C₂².₃₁(C₂×C₂²⋊C₄), SmallGroup(128,516)

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

Derived series

C₁ — C₂³ — C₄.C₂²≀C₂

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — Q₈×C₂³ — C₄.C₂²≀C₂

Lower central

C₁ — C₂ — C₂³ — C₄.C₂²≀C₂

Upper central

C₁ — C₂² — C₂³×C₄ — C₄.C₂²≀C₂

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₄.C₂²≀C₂

Generators and relations for C₄.C₂²≀C₂
G = < a,b,c,d,e,f | a⁴=c²=d²=e²=1, b²=a², f²=a, bab^-1=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=a²bd, cd=dc, fcf^-1=ce=ec, de=ed, fdf^-1=a²d, ef=fe >

Subgroups: 500 in 282 conjugacy classes, 72 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂²⋊C₈, C₄.₁₀D₄, C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂⁴.₄C₄, C₂×C₄.₁₀D₄, Q₈×C₂³, C₄.C₂²≀C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄.₁₀D₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂⁴⋊₃C₄, C₂×C₄.₁₀D₄, C₄.C₂²≀C₂

Smallest permutation representation of C₄.C₂²≀C₂
►On 32 points

Generators in S₃₂

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

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

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

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

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

(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)

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4N	8A	···	8H
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	2	2	2	2	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	2	2	4
type	+	+	+	+			+	+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	C₄.₁₀D₄
kernel	C₄.C₂²≀C₂	C₂⁴.₄C₄	C₂×C₄.₁₀D₄	Q₈×C₂³	C₂³×C₄	C₂²×Q₈	C₂²×C₄	C₂×Q₈	C₂²
# reps	1	2	4	1	4	4	4	8	4

Matrix representation of C₄.C₂²≀C₂ ►in GL₆(𝔽₁₇)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	10	0	0
0	0	10	1	0	0
0	0	0	0	1	7
0	0	0	0	7	16

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

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

16	0	0	0	0	0
0	16	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

13	8	0	0	0	0
13	4	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	16	0	0	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,10,0,0,0,0,10,1,0,0,0,0,0,0,1,7,0,0,0,0,7,16],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₄.C₂²≀C₂ in GAP, Magma, Sage, TeX

C_4.C_2^2\wr C_2

% in TeX

G:=Group("C4.C2^2wrC2");

// GroupNames label

G:=SmallGroup(128,516);

// by ID

G=gap.SmallGroup(128,516);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,2028,124]);

// Polycyclic

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

// generators/relations

G = C4.C22≀C2 order 128 = 27

2nd non-split extension by C4 of C22≀C2 acting via C22≀C2/C2×D4=C2

G = C₄.C₂²≀C₂ order 128 = 2⁷

2^nd non-split extension by C₄ of C₂²≀C₂ acting via C₂²≀C₂/C₂×D₄=C₂