C2^2.78C2^5 - GroupNames

G = C₂².₇₈C₂⁵ order 128 = 2⁷

59^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₇₈C₂⁵, C₄².₈₀C₂³, C₂³.₁₃₀C₂⁴, C₂⁴.₅₀₃C₂³, C₂²⋊₂2_-¹⁺⁴, C₄○D₄⋊₁₈D₄, (D₄×Q₈)⋊₁₆C₂, Q₈ ○(C₄⋊D₄), D₄ ○(C₂²⋊Q₈), D₄.₅₈(C₂×D₄), C₄⋊Q₈⋊₃₀C₂², Q₈.₅₉(C₂×D₄), D₄⋊₅D₄⋊₁₅C₂, D₄⋊₆D₄⋊₁₈C₂, Q₈⋊₅D₄⋊₁₅C₂, (C₄×D₄)⋊₃₆C₂², (C₂×C₄).₇₁C₂⁴, (C₄×Q₈)⋊₃₈C₂², C₂.₃₀(D₄×C₂³), C₄⋊C₄.₂₉₂C₂³, C₄⋊D₄⋊₂₁C₂², C₄.₁₁₉(C₂²×D₄), C₂²⋊Q₈⋊₉₅C₂², (C₂×D₄).₂₉₉C₂³, C₄.₄D₄⋊₂₄C₂², C₂²⋊C₄.₁₈C₂³, (C₂×2_-¹⁺⁴)⋊₇C₂, (C₂×Q₈).₄₄₃C₂³, (C₂²×Q₈)⋊₃₁C₂², C₂².₁₂(C₂²×D₄), C₂².₁₉C₂⁴⋊₂₅C₂, C₄²⋊C₂⋊₃₃C₂², C₂²≀C₂.₂₆C₂², (C₂³×C₄).₆₀₅C₂², (C₂²×C₄).₃₅₂C₂³, C₂².D₄⋊₃C₂², C₂.₁₉(C₂×2_-¹⁺⁴), C₂.₁₇(C₂.C₂⁵), (C₂²×D₄).₅₉₈C₂², C₂³.₃₈C₂³⋊₂₀C₂, C₂³.₃₃C₂³⋊₁₆C₂, C₂².₃₁C₂⁴⋊₁₂C₂, (C₂×C₄⋊C₄)⋊₆₉C₂², (C₂×C₄).₆₆₆(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₇₅C₂, (C₂²×C₄○D₄)⋊₂₄C₂, (C₂×C₄○D₄)⋊₂₆C₂², (C₂×C₂²⋊C₄).₃₇₉C₂², SmallGroup(128,2221)

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

Derived series

C₁ — C₂² — C₂².₇₈C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂².₇₈C₂⁵

Lower central

C₁ — C₂² — C₂².₇₈C₂⁵

Upper central

C₁ — C₂² — C₂².₇₈C₂⁵

Jennings

C₁ — C₂² — C₂².₇₈C₂⁵

Generators and relations for C₂².₇₈C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=d²=e²=f²=1, c²=g²=a, ab=ba, dcd=gcg^-1=ac=ca, fdf=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: 1156 in 761 conjugacy classes, 430 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊Q₈, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂³.₃₃C₂³, C₂×C₂²⋊Q₈, C₂².₁₉C₂⁴, C₂³.₃₈C₂³, C₂².₃₁C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, Q₈⋊₅D₄, D₄×Q₈, C₂²×C₄○D₄, C₂×2_-¹⁺⁴, C₂².₇₈C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, 2_-¹⁺⁴, C₂⁵, D₄×C₂³, C₂×2_-¹⁺⁴, C₂.C₂⁵, C₂².₇₈C₂⁵

Smallest permutation representation of C₂².₇₈C₂⁵
►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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2K	2L	···	2P	4A	···	4J	4K	···	4AA
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

type

image

C₁

C₂

D₄

2_-¹⁺⁴

C₂.C₂⁵

kernel

C₂².₇₈C₂⁵

C₂³.₃₃C₂³

C₂×C₂²⋊Q₈

C₂².₁₉C₂⁴

C₂³.₃₈C₂³

C₂².₃₁C₂⁴

D₄⋊₅D₄

D₄⋊₆D₄

Q₈⋊₅D₄

D₄×Q₈

C₂²×C₄○D₄

C₂×2_-¹⁺⁴

C₄○D₄

C₂²

C₂

# reps

Matrix representation of C₂².₇₈C₂⁵ ►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

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

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

1	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	1	0
0	0	0	0	0	1

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

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

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

C₂².₇₈C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{78}C_2^5

% in TeX

G:=Group("C2^2.78C2^5");

// GroupNames label

G:=SmallGroup(128,2221);

// by ID

G=gap.SmallGroup(128,2221);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,570,1684,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=e^2=f^2=1,c^2=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=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 = C22.78C25 order 128 = 27

59th central stem extension by C22 of C25

G = C₂².₇₈C₂⁵ order 128 = 2⁷

59^th central stem extension by C₂² of C₂⁵