C2^2.87C2^5 - GroupNames

G = C₂².₈₇C₂⁵ order 128 = 2⁷

68^th central stem extension by C₂² of C₂⁵

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C₄⋊₂2₊¹⁺⁴, C₂³.₄₃C₂⁴, C₂².₈₇C₂⁵, C₄².₅₇₅C₂³, C₂⁴.₁₃₅C₂³, D₄²⋊₁₄C₂, Q₈ ○(C₄⋊Q₈), C₄○D₄⋊₁₀D₄, D₄⋊₁₆(C₂×D₄), D₄ ○(C₄⋊₁D₄), Q₈⋊₁₄(C₂×D₄), Q₈⋊₆D₄⋊₁₈C₂, (C₄×D₄)⋊₄₄C₂², (C₂×C₄).₇₇C₂⁴, C₄⋊Q₈⋊₁₀₈C₂², (C₄×Q₈)⋊₉₉C₂², C₂.₃₃(D₄×C₂³), C₂²≀C₂⋊₈C₂², C₄⋊₁D₄⋊₅₁C₂², C₄⋊C₄.₄₈₉C₂³, C₄⋊D₄⋊₂₇C₂², (C₂×C₄²)⋊₅₉C₂², C₄.₁₂₂(C₂²×D₄), (C₂×D₄).₄₇₀C₂³, C₄.₄D₄⋊₈₂C₂², (C₂²×D₄)⋊₃₇C₂², C₂²⋊C₄.₂₂C₂³, (C₂×Q₈).₄₄₇C₂³, C₂².₁₄(C₂²×D₄), C₂².₂₉C₂⁴⋊₂₂C₂, (C₂×2₊¹⁺⁴)⋊₁₁C₂, C₄²⋊C₂⋊₁₀₀C₂², C₂.₃₂(C₂×2₊¹⁺⁴), C₂².₂₆C₂⁴⋊₃₆C₂, (C₂²×C₄).₁₂₀₈C₂³, (C₂×C₄)⋊₅(C₂×D₄), (C₄×C₄○D₄)⋊₂₆C₂, (C₂×D₄)○(C₄⋊₁D₄), (C₂×C₄⋊₁D₄)⋊₂₆C₂, (C₂×C₄○D₄)⋊₃₀C₂², SmallGroup(128,2230)

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

Derived series

C₁ — C₂² — C₂².₈₇C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₄×C₄○D₄ — C₂².₈₇C₂⁵

Lower central

C₁ — C₂² — C₂².₈₇C₂⁵

Upper central

C₁ — C₂² — C₂².₈₇C₂⁵

Jennings

C₁ — C₂² — C₂².₈₇C₂⁵

Generators and relations for C₂².₈₇C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=b, g²=a, ab=ba, dcd=gcg^-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece^-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1628 in 874 conjugacy classes, 432 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂×C₄², C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₄×C₄○D₄, C₂×C₄⋊₁D₄, C₂².₂₆C₂⁴, C₂².₂₉C₂⁴, D₄², Q₈⋊₆D₄, C₂×2₊¹⁺⁴, C₂².₈₇C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, 2₊¹⁺⁴, C₂⁵, D₄×C₂³, C₂×2₊¹⁺⁴, C₂².₈₇C₂⁵

Smallest permutation representation of C₂².₈₇C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2U	4A	···	4L	4M	···	4V
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	2₊¹⁺⁴
kernel	C₂².₈₇C₂⁵	C₄×C₄○D₄	C₂×C₄⋊₁D₄	C₂².₂₆C₂⁴	C₂².₂₉C₂⁴	D₄²	Q₈⋊₆D₄	C₂×2₊¹⁺⁴	C₄○D₄	C₄
# reps	1	1	3	3	6	12	4	2	8	4

Matrix representation of C₂².₈₇C₂⁵ ►in GL₆(ℤ)

1	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	0
0	0	0	0	0	-1

-1	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	0
0	0	0	0	0	1

-1	-2	0	0	0	0
0	1	0	0	0	0
0	0	0	-1	0	0
0	0	-1	0	0	0
0	0	0	0	0	-1
0	0	0	0	-1	0

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	0	0	0	1
0	0	0	0	-1	0
0	0	0	-1	0	0
0	0	1	0	0	0

-1	-2	0	0	0	0
1	1	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	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	0	1	0	0
0	0	-1	0	0	0
0	0	0	0	0	1
0	0	0	0	-1	0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1,0,0,0],[-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0] >;

C₂².₈₇C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{87}C_2^5

% in TeX

G:=Group("C2^2.87C2^5");

// GroupNames label

G:=SmallGroup(128,2230);

// by ID

G=gap.SmallGroup(128,2230);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,570,1684,102]);

// Polycyclic

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

// generators/relations

G = C22.87C25 order 128 = 27

68th central stem extension by C22 of C25

G = C₂².₈₇C₂⁵ order 128 = 2⁷

68^th central stem extension by C₂² of C₂⁵