C2^2.75C2^5 - GroupNames

G = C₂².₇₅C₂⁵ order 128 = 2⁷

56^th central stem extension by C₂² of C₂⁵

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

Aliases: C₄².₇₇C₂³, C₂².₇₅C₂⁵, C₂³.₃₅C₂⁴, C₂⁴.₆₁₉C₂³, C₂²⋊₁2_-¹⁺⁴, Q₈ ○C₂²≀C₂, (D₄×Q₈)⋊₁₅C₂, (C₂×Q₈)⋊₄₀D₄, C₄⋊Q₈⋊₂₉C₂², Q₈.₅₇(C₂×D₄), Q₈⋊₅D₄⋊₁₂C₂, (C₂×C₄).₆₉C₂⁴, (C₄×Q₈)⋊₃₅C₂², (Q₈×C₂³)⋊₁₂C₂, C₂.₂₇(D₄×C₂³), C₄⋊C₄.₄₈₂C₂³, C₄.₁₁₆(C₂²×D₄), C₂²⋊Q₈⋊₂₃C₂², (C₄×D₄).₂₂₉C₂², (C₂×D₄).₄₆₄C₂³, C₄.₄D₄⋊₂₁C₂², C₂²⋊C₄.₁₆C₂³, (C₂×2_-¹⁺⁴)⋊₅C₂, (C₂×Q₈).₄₄₀C₂³, (C₂²×Q₈)⋊₂₉C₂², C₂².₅₀(C₂²×D₄), C₂².₁₉C₂⁴⋊₂₂C₂, C₂²≀C₂.₃₅C₂², C₄⋊D₄.₂₂₃C₂², (C₂³×C₄).₆₀₃C₂², C₂.₁₈(C₂×2_-¹⁺⁴), (C₂²×C₄).₁₂₀₆C₂³, C₄²⋊C₂.₂₂₃C₂², C₂³.₃₈C₂³⋊₁₉C₂, C₂³.₃₂C₂³⋊₁₁C₂, C₂².D₄.₂₉C₂², (C₂×Q₈)○C₂²≀C₂, (C₂×C₄).₆₆₄(C₂×D₄), (C₂×C₄○D₄)⋊₂₃C₂², SmallGroup(128,2218)

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

Derived series

C₁ — C₂² — C₂².₇₅C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — Q₈×C₂³ — C₂².₇₅C₂⁵

Lower central

C₁ — C₂² — C₂².₇₅C₂⁵

Upper central

C₁ — C₂² — C₂².₇₅C₂⁵

Jennings

C₁ — C₂² — C₂².₇₅C₂⁵

Generators and relations for C₂².₇₅C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=g²=1, d²=f²=a, ab=ba, dcd^-1=gcg=ac=ca, fdf^-1=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1068 in 746 conjugacy classes, 432 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, 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⋊Q₈, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, C₂³.₃₂C₂³, C₂².₁₉C₂⁴, C₂³.₃₈C₂³, Q₈⋊₅D₄, D₄×Q₈, Q₈×C₂³, C₂×2_-¹⁺⁴, C₂².₇₅C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, 2_-¹⁺⁴, C₂⁵, D₄×C₂³, C₂×2_-¹⁺⁴, C₂².₇₅C₂⁵

Smallest permutation representation of C₂².₇₅C₂⁵
►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 15)(2 16)(3 13)(4 14)(5 19)(6 20)(7 17)(8 18)(9 23)(10 24)(11 21)(12 22)(25 29)(26 30)(27 31)(28 32)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	···	4L	4M	···	4AD
order	1	2	2	2	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	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	C₂².₇₅C₂⁵	C₂³.₃₂C₂³	C₂².₁₉C₂⁴	C₂³.₃₈C₂³	Q₈⋊₅D₄	D₄×Q₈	Q₈×C₂³	C₂×2_-¹⁺⁴	C₂×Q₈	C₂²
# reps	1	1	6	6	8	8	1	1	8	4

Matrix representation of C₂².₇₅C₂⁵ ►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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

4	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	1	0
0	0	2	2	0	1

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

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,0,0,0,0,3,0,4,0,0,1,1,0,3,0,0,0,3,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,1,0,0,0,0,2,2,0,0,0,0,0,3],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,2,0,0,0,4,0,2,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,3,0,0,0,1,0,3,3,0,0,0,0,1,1,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,2,0,0,0,4,0,2,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂².₇₅C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{75}C_2^5

% in TeX

G:=Group("C2^2.75C2^5");

// GroupNames label

G:=SmallGroup(128,2218);

// by ID

G=gap.SmallGroup(128,2218);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=g^2=1,d^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=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 = C22.75C25 order 128 = 27

56th central stem extension by C22 of C25

G = C₂².₇₅C₂⁵ order 128 = 2⁷

56^th central stem extension by C₂² of C₂⁵