C2^4.176C2^3 - GroupNames

G = C₂⁴.₁₇₆C₂³ order 128 = 2⁷

16^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₁₇₆C₂³, C₂²⋊C₄⋊₉Q₈, (C₂²×Q₈)⋊₇C₄, C₂³.₆(C₂×Q₈), C₂².₁₃(C₄×Q₈), C₄.₁₅(C₂³⋊C₄), C₂³.₉(C₄○D₄), C₂³.₅₇₆(C₂×D₄), (C₂²×C₄).₃₀₆D₄, C₂².₁₂(C₄⋊Q₈), C₂³.₉D₄.₆C₂, C₂³.₂₀₀(C₂²×C₄), (C₂³×C₄).₂₆₆C₂², C₂².₃₀(C₂²⋊Q₈), C₂³.₇Q₈.₁₇C₂, C₂².₂₂(C₄.₄D₄), C₂.₂₉(C₂³.C₂³), C₂.₁₈(C₂³.₆₇C₂³), (C₂×C₄⋊C₄)⋊₁₅C₄, C₂.₂₉(C₂×C₂³⋊C₄), (C₄×C₂²⋊C₄).₁₇C₂, (C₂²×C₄).₂₆(C₂×C₄), (C₂×C₂²⋊Q₈).₁₁C₂, (C₂×C₄).₂₀₉(C₂²⋊C₄), (C₂×C₂²⋊C₄).₁₆C₂², C₂².₃₀₅(C₂×C₂²⋊C₄), SmallGroup(128,728)

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

Derived series

C₁ — C₂³ — C₂⁴.₁₇₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₁₇₆C₂³

Lower central

C₁ — C₂ — C₂³ — C₂⁴.₁₇₆C₂³

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₇₆C₂³

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₁₇₆C₂³

Generators and relations for C₂⁴.₁₇₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=abc, g²=b, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, ag=ga, bc=cb, bd=db, geg^-1=be=eb, bf=fb, bg=gb, fcf^-1=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef^-1=ae, fg=gf >

Subgroups: 364 in 162 conjugacy classes, 58 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂³.₉D₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂×C₂²⋊Q₈, C₂⁴.₁₇₆C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₄×Q₈, C₂²⋊Q₈, C₄.₄D₄, C₄⋊Q₈, C₂³.₆₇C₂³, C₂×C₂³⋊C₄, C₂³.C₂³, C₂⁴.₁₇₆C₂³

Smallest permutation representation of C₂⁴.₁₇₆C₂³
►On 32 points

Generators in S₃₂

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

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

(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4N	4O	···	4V
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	4	···	4
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	1	2	2	2	4	4
type	+	+	+	+	+			-	+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	Q₈	D₄	C₄○D₄	C₂³⋊C₄	C₂³.C₂³
kernel	C₂⁴.₁₇₆C₂³	C₂³.₉D₄	C₄×C₂²⋊C₄	C₂³.₇Q₈	C₂×C₂²⋊Q₈	C₂×C₄⋊C₄	C₂²×Q₈	C₂²⋊C₄	C₂²×C₄	C₂³	C₄	C₂
# reps	1	4	1	1	1	6	2	4	4	4	2	2

Matrix representation of C₂⁴.₁₇₆C₂³ ►in GL₆(𝔽₅)

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

4	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
0	0	0	0	0	4

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

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

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

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

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

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

C₂⁴.₁₇₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{176}C_2^3

% in TeX

G:=Group("C2^4.176C2^3");

// GroupNames label

G:=SmallGroup(128,728);

// by ID

G=gap.SmallGroup(128,728);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,1018,2028]);

// Polycyclic

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

// generators/relations

G = C24.176C23 order 128 = 27

16th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₁₇₆C₂³ order 128 = 2⁷

16^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²