C2^2.76C2^5 - GroupNames

G = C₂².₇₆C₂⁵ order 128 = 2⁷

57^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄².₇₈C₂³, C₂².₇₆C₂⁵, C₂³.₃₆C₂⁴, C₂⁴.₅₀₂C₂³, (C₂×Q₈)⋊₄₁D₄, Q₈.₅₈(C₂×D₄), Q₈⋊₅D₄⋊₁₃C₂, Q₈⋊₆D₄⋊₁₅C₂, (C₄×D₄)⋊₃₄C₂², (C₄×Q₈)⋊₃₆C₂², C₂.₂₈(D₄×C₂³), C₄⋊₁D₄⋊₁₆C₂², C₄⋊D₄⋊₈₀C₂², C₄⋊C₄.₄₈₃C₂³, (C₂×C₄).₁₆₉C₂⁴, C₄.₁₁₇(C₂²×D₄), C₂²⋊Q₈⋊₉₄C₂², Q₈ ○(C₂².D₄), (C₂×D₄).₂₉₈C₂³, C₄.₄D₄⋊₂₂C₂², C₂²⋊C₄.₉₅C₂³, (C₂×2_-¹⁺⁴)⋊₆C₂, (C₂×Q₈).₄₄₁C₂³, (C₂²×Q₈)⋊₃₀C₂², C₂².₅₁(C₂²×D₄), C₂².₂₉C₂⁴⋊₁₉C₂, C₂².₁₉C₂⁴⋊₂₃C₂, C₂²≀C₂.₂₅C₂², (C₂³×C₄).₆₀₄C₂², C₂.₁₅(C₂.C₂⁵), (C₂²×C₄).₁₂₀₇C₂³, (C₂²×D₄).₅₉₇C₂², C₂³.₃₂C₂³⋊₁₂C₂, C₄²⋊C₂.₂₂₄C₂², C₂².D₄.₄₂C₂², (C₂×C₄).₆₆₅(C₂×D₄), (C₂²×C₄○D₄)⋊₂₂C₂, (C₂×C₄○D₄)⋊₂₄C₂², SmallGroup(128,2219)

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

Derived series

C₁ — C₂² — C₂².₇₆C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂².₇₆C₂⁵

Lower central

C₁ — C₂² — C₂².₇₆C₂⁵

Upper central

C₁ — C₂² — C₂².₇₆C₂⁵

Jennings

C₁ — C₂² — C₂².₇₆C₂⁵

Generators and relations for C₂².₇₆C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=e²=f²=a, ab=ba, dcd^-1=gcg=ac=ca, fdf^-1=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: 1212 in 768 conjugacy classes, 428 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊₁D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂³.₃₂C₂³, C₂².₁₉C₂⁴, C₂².₂₉C₂⁴, Q₈⋊₅D₄, Q₈⋊₆D₄, C₂²×C₄○D₄, C₂×2_-¹⁺⁴, C₂².₇₆C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₄×C₂³, C₂.C₂⁵, C₂².₇₆C₂⁵

Smallest permutation representation of C₂².₇₆C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2O	4A	···	4P	4Q	···	4AB
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	C₂.C₂⁵
kernel	C₂².₇₆C₂⁵	C₂³.₃₂C₂³	C₂².₁₉C₂⁴	C₂².₂₉C₂⁴	Q₈⋊₅D₄	Q₈⋊₆D₄	C₂²×C₄○D₄	C₂×2_-¹⁺⁴	C₂×Q₈	C₂
# reps	1	1	6	6	8	8	1	1	8	4

Matrix representation of C₂².₇₆C₂⁵ ►in GL₆(𝔽₅)

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	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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	4	1	1	4
0	0	1	0	0	0
0	0	0	0	0	4

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

1	0	0	0	0	0
0	4	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

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

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	4	0
0	0	3	2	0	4

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

C₂².₇₆C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{76}C_2^5

% in TeX

G:=Group("C2^2.76C2^5");

// GroupNames label

G:=SmallGroup(128,2219);

// by ID

G=gap.SmallGroup(128,2219);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,184,570,136,1684]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=f^2=a,a*b=b*a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f^-1=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 = C22.76C25 order 128 = 27

57th central stem extension by C22 of C25

G = C₂².₇₆C₂⁵ order 128 = 2⁷

57^th central stem extension by C₂² of C₂⁵