C2^4.175C2^3 - GroupNames

G = C₂⁴.₁₇₅C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₇₅C₂³, C₄⋊₁(C₂³⋊C₄), C₂²⋊C₄⋊₂₀D₄, (C₂²×D₄)⋊₈C₄, C₂⁴.₁₀(C₂×C₄), C₂².₅₇(C₄×D₄), C₂³.₈(C₄○D₄), C₂³.₅₇₁(C₂×D₄), (C₂²×C₄).₃₀₀D₄, C₂³.₇Q₈⋊₈C₂, C₂².₄(C₄⋊₁D₄), C₂².₁₈(C₄⋊D₄), C₂³.₁₉₆(C₂²×C₄), (C₂³×C₄).₂₆₅C₂², (C₂²×D₄).₄₅C₂², C₂².₃₄(C₄.₄D₄), C₂.₂₈(C₂³.C₂³), C₂.₁₆(C₂⁴.₃C₂²), (C₂×C₄⋊C₄)⋊₁₄C₄, (C₂×C₂³⋊C₄)⋊₇C₂, (C₂×C₂²⋊C₄)⋊₆C₄, (C₄×C₂²⋊C₄)⋊₂₄C₂, C₂.₂₈(C₂×C₂³⋊C₄), (C₂×C₄⋊D₄).₁₁C₂, (C₂²×C₄).₂₃(C₂×C₄), (C₂×C₄).₂₀₁(C₂²⋊C₄), (C₂×C₂²⋊C₄).₁₅C₂², C₂².₂₈₆(C₂×C₂²⋊C₄), SmallGroup(128,696)

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

Derived series

C₁ — C₂³ — C₂⁴.₁₇₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₁₇₅C₂³

Lower central

C₁ — C₂ — C₂³ — C₂⁴.₁₇₅C₂³

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₇₅C₂³

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₁₇₅C₂³

Generators and relations for C₂⁴.₁₇₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=ca=ac, g²=b, ab=ba, 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: 508 in 204 conjugacy classes, 58 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂³⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂×C₂³⋊C₄, C₂×C₄⋊D₄, C₂⁴.₁₇₅C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₄×D₄, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₂⁴.₃C₂², C₂×C₂³⋊C₄, C₂³.C₂³, C₂⁴.₁₇₅C₂³

Smallest permutation representation of C₂⁴.₁₇₅C₂³
►On 32 points

Generators in S₃₂

(2 28)(4 26)(6 17)(8 19)(10 14)(12 16)(22 30)(24 32)

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	4A	4B	4C	4D	4E	···	4N	4O	···	4T
order	1	2	2	2	2	···	2	2	2	4	4	4	4	4	···	4	4	···	4
size	1	1	1	1	2	···	2	8	8	2	2	2	2	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+				+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	C₄○D₄	C₂³⋊C₄	C₂³.C₂³
kernel	C₂⁴.₁₇₅C₂³	C₄×C₂²⋊C₄	C₂³.₇Q₈	C₂×C₂³⋊C₄	C₂×C₄⋊D₄	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₂²×D₄	C₂²⋊C₄	C₂²×C₄	C₂³	C₄	C₂
# reps	1	1	1	4	1	4	2	2	4	4	4	2	2

Matrix representation of C₂⁴.₁₇₅C₂³ ►in GL₆(𝔽₅)

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	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	1	0
0	0	0	0	0	1

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

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

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

0	4	0	0	0	0
1	0	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

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

C₂⁴.₁₇₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{175}C_2^3

% in TeX

G:=Group("C2^4.175C2^3");

// GroupNames label

G:=SmallGroup(128,696);

// by ID

G=gap.SmallGroup(128,696);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2028]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=c*a=a*c,g^2=b,a*b=b*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 = C24.175C23 order 128 = 27

15th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₁₇₅C₂³ order 128 = 2⁷

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