C2^4.162C2^3 - GroupNames

G = C₂⁴.₁₆₂C₂³ order 128 = 2⁷

2^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₁₆₂C₂³, (C₂×C₄²)⋊₅C₄, (C₂×C₄).₆₈C₄², C₄²⋊C₂⋊₁₇C₄, C₂³.₅₁(C₂×Q₈), (C₂²×C₄).₃₈Q₈, C₄ ○(C₂³.₉D₄), C₂³.₅₃₉(C₂×D₄), (C₂²×C₄).₇₅₅D₄, C₂².₅(C₂×C₄²), C₂³.₉D₄.₉C₂, (C₂³×C₄).₂₂₀C₂², C₂³.₁₇₅(C₂²×C₄), C₄.₂₁(C₂.C₄²), C₂.₄(C₂³.C₂³), (C₂×C₄⋊C₄)⋊₂₄C₄, C₂².₁₃(C₂×C₄⋊C₄), (C₂×C₄).₁₂₅(C₄⋊C₄), C₂²⋊C₄.₄₇(C₂×C₄), (C₂²×C₄).₄₆(C₂×C₄), (C₂×C₄²⋊C₂).₉C₂, (C₂×C₄).₁₁₃(C₂²⋊C₄), C₂².₁₀₈(C₂×C₂²⋊C₄), C₂.₁₇(C₂×C₂.C₄²), (C₂×C₂²⋊C₄).₄₀₇C₂², SmallGroup(128,472)

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

Derived series

C₁ — C₂² — C₂⁴.₁₆₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₄²⋊C₂ — C₂⁴.₁₆₂C₂³

Lower central

C₁ — C₂ — C₂² — C₂⁴.₁₆₂C₂³

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂⁴.₁₆₂C₂³

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₁₆₂C₂³

Generators and relations for C₂⁴.₁₆₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=b, f²=c, g²=d, ab=ba, ac=ca, eae^-1=ad=da, af=fa, ag=ga, bc=cb, fbf^-1=bd=db, be=eb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fef^-1=ade, eg=ge, fg=gf >

Subgroups: 372 in 210 conjugacy classes, 108 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂³×C₄, C₂³.₉D₄, C₂×C₄²⋊C₂, C₂×C₄²⋊C₂, C₂⁴.₁₆₂C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₂.C₄², C₂³.C₂³, C₂⁴.₁₆₂C₂³

Smallest permutation representation of C₂⁴.₁₆₂C₂³
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	2	2	4
type	+	+	+				+	-
image	C₁	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	C₂³.C₂³
kernel	C₂⁴.₁₆₂C₂³	C₂³.₉D₄	C₂×C₄²⋊C₂	C₂×C₄²	C₂×C₄⋊C₄	C₄²⋊C₂	C₂²×C₄	C₂²×C₄	C₂
# reps	1	4	3	4	4	16	6	2	4

Matrix representation of C₂⁴.₁₆₂C₂³ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	0	0	0	4
0	0	0	0	4	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

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	1	0
0	0	0	0	0	1

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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	3

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

C₂⁴.₁₆₂C₂³ in GAP, Magma, Sage, TeX

C_2^4._{162}C_2^3

% in TeX

G:=Group("C2^4.162C2^3");

// GroupNames label

G:=SmallGroup(128,472);

// by ID

G=gap.SmallGroup(128,472);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,352,2019,1411]);

// Polycyclic

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

// generators/relations

G = C24.162C23 order 128 = 27

2nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₁₆₂C₂³ order 128 = 2⁷

2^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²