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^[×3], C₂^[×6], C₄^[×2], C₄^[×13], C₂²^[×3], C₂²^[×4], C₂²^[×10], C₂×C₄^[×4], C₂×C₄^[×31], Q₈^[×4], C₂³^[×3], C₂³^[×4], C₂³^[×2], C₄²^[×2], C₂²⋊C₄^[×4], C₂²⋊C₄^[×10], C₄⋊C₄^[×8], C₂²×C₄^[×6], C₂²×C₄^[×4], C₂²×C₄^[×6], C₂×Q₈^[×4], C₂⁴, C₂.C₄²^[×2], C₂×C₄², C₂×C₂²⋊C₄^[×6], C₂×C₄⋊C₄, C₂×C₄⋊C₄^[×2], C₂×C₄⋊C₄, C₂²⋊Q₈^[×4], C₂³×C₄, C₂²×Q₈, C₂³.₉D₄^[×4], C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂×C₂²⋊Q₈, C₂⁴.₁₇₆C₂³
Quotients: C₁, C₂^[×7], C₄^[×4], C₂²^[×7], C₂×C₄^[×6], D₄^[×4], Q₈^[×4], C₂³, C₂²⋊C₄^[×4], C₂²×C₄, C₂×D₄^[×2], C₂×Q₈^[×2], C₄○D₄^[×2], C₂³⋊C₄^[×2], C₂×C₂²⋊C₄, C₄×Q₈^[×2], C₂²⋊Q₈^[×2], 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 7)(2 15)(3 9)(4 13)(5 8)(6 16)(10 11)(12 14)(17 28)(18 27)(19 26)(20 25)(21 31)(22 30)(23 29)(24 32)

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

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

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

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

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

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

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

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