C2^3.643C2^4 - GroupNames

G = C₂³.₆₄₃C₂⁴ order 128 = 2⁷

360^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₄₃C₂⁴, C₂⁴.₄₂₉C₂³, C₂².₄₁₆2₊¹⁺⁴, C₂².₃₁₅2_-¹⁺⁴, (C₂×C₄²).₉₁C₂², C₂³.Q₈⋊₇₇C₂, C₂³.₄Q₈⋊₅₇C₂, C₂³.₁₈₅(C₄○D₄), C₂³.₃₄D₄⋊₅₃C₂, (C₂²×C₄).₂₀₄C₂³, (C₂³×C₄).₁₅₈C₂², C₂³.₈Q₈⋊₁₂₃C₂, C₂³.₇Q₈⋊₁₀₂C₂, C₂³.₂₃D₄.₆₃C₂, C₂³.₁₀D₄.₅₅C₂, (C₂²×D₄).₂₆₃C₂², C₂⁴.C₂²⋊₁₅₅C₂, C₂³.₈₃C₂³⋊₉₅C₂, C₂.₈₀(C₂².₃₂C₂⁴), C₂³.₆₃C₂³⋊₁₅₉C₂, C₂.₉₅(C₂².₄₅C₂⁴), C₂.C₄².₃₄₇C₂², C₂.₅₁(C₂².₃₄C₂⁴), C₂.₉₂(C₂².₄₆C₂⁴), C₂.₂₈(C₂².₅₆C₂⁴), C₂.₈₅(C₂².₃₃C₂⁴), (C₂×C₄).₄₄₄(C₄○D₄), (C₂×C₄⋊C₄).₄₅₄C₂², C₂².₅₀₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₆₁C₂², SmallGroup(128,1475)

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

Derived series

C₁ — C₂³ — C₂³.₆₄₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂³.₆₄₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₄₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₄₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₄₃C₂⁴

Generators and relations for C₂³.₆₄₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=cb=bc, e²=b, f²=a, ab=ba, ac=ca, ede^-1=ad=da, geg=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 468 in 224 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₄₃C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₄₃C₂⁴

Smallest permutation representation of C₂³.₆₄₃C₂⁴
►On 64 points

Generators in S₆₄

(1 58)(2 59)(3 60)(4 57)(5 56)(6 53)(7 54)(8 55)(9 46)(10 47)(11 48)(12 45)(13 50)(14 51)(15 52)(16 49)(17 41)(18 42)(19 43)(20 44)(21 39)(22 40)(23 37)(24 38)(25 35)(26 36)(27 33)(28 34)(29 61)(30 62)(31 63)(32 64)

(1 61)(2 62)(3 63)(4 64)(5 27)(6 28)(7 25)(8 26)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(21 52)(22 49)(23 50)(24 51)(29 58)(30 59)(31 60)(32 57)(33 56)(34 53)(35 54)(36 55)

(1 63)(2 64)(3 61)(4 62)(5 25)(6 26)(7 27)(8 28)(9 43)(10 44)(11 41)(12 42)(13 39)(14 40)(15 37)(16 38)(17 48)(18 45)(19 46)(20 47)(21 50)(22 51)(23 52)(24 49)(29 60)(30 57)(31 58)(32 59)(33 54)(34 55)(35 56)(36 53)

(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 17 61 46)(2 42 62 10)(3 19 63 48)(4 44 64 12)(5 49 27 22)(6 13 28 37)(7 51 25 24)(8 15 26 39)(9 58 41 29)(11 60 43 31)(14 35 38 54)(16 33 40 56)(18 30 47 59)(20 32 45 57)(21 55 52 36)(23 53 50 34)

(1 51 58 14)(2 21 59 39)(3 49 60 16)(4 23 57 37)(5 41 56 17)(6 10 53 47)(7 43 54 19)(8 12 55 45)(9 33 46 27)(11 35 48 25)(13 64 50 32)(15 62 52 30)(18 28 42 34)(20 26 44 36)(22 31 40 63)(24 29 38 61)

(1 34)(2 7)(3 36)(4 5)(6 29)(8 31)(9 50)(10 38)(11 52)(12 40)(13 46)(14 42)(15 48)(16 44)(17 37)(18 51)(19 39)(20 49)(21 43)(22 45)(23 41)(24 47)(25 62)(26 60)(27 64)(28 58)(30 35)(32 33)(53 61)(54 59)(55 63)(56 57)

G:=sub<Sym(64)| (1,58)(2,59)(3,60)(4,57)(5,56)(6,53)(7,54)(8,55)(9,46)(10,47)(11,48)(12,45)(13,50)(14,51)(15,52)(16,49)(17,41)(18,42)(19,43)(20,44)(21,39)(22,40)(23,37)(24,38)(25,35)(26,36)(27,33)(28,34)(29,61)(30,62)(31,63)(32,64), (1,61)(2,62)(3,63)(4,64)(5,27)(6,28)(7,25)(8,26)(9,41)(10,42)(11,43)(12,44)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(21,52)(22,49)(23,50)(24,51)(29,58)(30,59)(31,60)(32,57)(33,56)(34,53)(35,54)(36,55), (1,63)(2,64)(3,61)(4,62)(5,25)(6,26)(7,27)(8,28)(9,43)(10,44)(11,41)(12,42)(13,39)(14,40)(15,37)(16,38)(17,48)(18,45)(19,46)(20,47)(21,50)(22,51)(23,52)(24,49)(29,60)(30,57)(31,58)(32,59)(33,54)(34,55)(35,56)(36,53), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,17,61,46)(2,42,62,10)(3,19,63,48)(4,44,64,12)(5,49,27,22)(6,13,28,37)(7,51,25,24)(8,15,26,39)(9,58,41,29)(11,60,43,31)(14,35,38,54)(16,33,40,56)(18,30,47,59)(20,32,45,57)(21,55,52,36)(23,53,50,34), (1,51,58,14)(2,21,59,39)(3,49,60,16)(4,23,57,37)(5,41,56,17)(6,10,53,47)(7,43,54,19)(8,12,55,45)(9,33,46,27)(11,35,48,25)(13,64,50,32)(15,62,52,30)(18,28,42,34)(20,26,44,36)(22,31,40,63)(24,29,38,61), (1,34)(2,7)(3,36)(4,5)(6,29)(8,31)(9,50)(10,38)(11,52)(12,40)(13,46)(14,42)(15,48)(16,44)(17,37)(18,51)(19,39)(20,49)(21,43)(22,45)(23,41)(24,47)(25,62)(26,60)(27,64)(28,58)(30,35)(32,33)(53,61)(54,59)(55,63)(56,57)>;

G:=Group( (1,58)(2,59)(3,60)(4,57)(5,56)(6,53)(7,54)(8,55)(9,46)(10,47)(11,48)(12,45)(13,50)(14,51)(15,52)(16,49)(17,41)(18,42)(19,43)(20,44)(21,39)(22,40)(23,37)(24,38)(25,35)(26,36)(27,33)(28,34)(29,61)(30,62)(31,63)(32,64), (1,61)(2,62)(3,63)(4,64)(5,27)(6,28)(7,25)(8,26)(9,41)(10,42)(11,43)(12,44)(13,37)(14,38)(15,39)(16,40)(17,46)(18,47)(19,48)(20,45)(21,52)(22,49)(23,50)(24,51)(29,58)(30,59)(31,60)(32,57)(33,56)(34,53)(35,54)(36,55), (1,63)(2,64)(3,61)(4,62)(5,25)(6,26)(7,27)(8,28)(9,43)(10,44)(11,41)(12,42)(13,39)(14,40)(15,37)(16,38)(17,48)(18,45)(19,46)(20,47)(21,50)(22,51)(23,52)(24,49)(29,60)(30,57)(31,58)(32,59)(33,54)(34,55)(35,56)(36,53), (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,17,61,46)(2,42,62,10)(3,19,63,48)(4,44,64,12)(5,49,27,22)(6,13,28,37)(7,51,25,24)(8,15,26,39)(9,58,41,29)(11,60,43,31)(14,35,38,54)(16,33,40,56)(18,30,47,59)(20,32,45,57)(21,55,52,36)(23,53,50,34), (1,51,58,14)(2,21,59,39)(3,49,60,16)(4,23,57,37)(5,41,56,17)(6,10,53,47)(7,43,54,19)(8,12,55,45)(9,33,46,27)(11,35,48,25)(13,64,50,32)(15,62,52,30)(18,28,42,34)(20,26,44,36)(22,31,40,63)(24,29,38,61), (1,34)(2,7)(3,36)(4,5)(6,29)(8,31)(9,50)(10,38)(11,52)(12,40)(13,46)(14,42)(15,48)(16,44)(17,37)(18,51)(19,39)(20,49)(21,43)(22,45)(23,41)(24,47)(25,62)(26,60)(27,64)(28,58)(30,35)(32,33)(53,61)(54,59)(55,63)(56,57) );

G=PermutationGroup([[(1,58),(2,59),(3,60),(4,57),(5,56),(6,53),(7,54),(8,55),(9,46),(10,47),(11,48),(12,45),(13,50),(14,51),(15,52),(16,49),(17,41),(18,42),(19,43),(20,44),(21,39),(22,40),(23,37),(24,38),(25,35),(26,36),(27,33),(28,34),(29,61),(30,62),(31,63),(32,64)], [(1,61),(2,62),(3,63),(4,64),(5,27),(6,28),(7,25),(8,26),(9,41),(10,42),(11,43),(12,44),(13,37),(14,38),(15,39),(16,40),(17,46),(18,47),(19,48),(20,45),(21,52),(22,49),(23,50),(24,51),(29,58),(30,59),(31,60),(32,57),(33,56),(34,53),(35,54),(36,55)], [(1,63),(2,64),(3,61),(4,62),(5,25),(6,26),(7,27),(8,28),(9,43),(10,44),(11,41),(12,42),(13,39),(14,40),(15,37),(16,38),(17,48),(18,45),(19,46),(20,47),(21,50),(22,51),(23,52),(24,49),(29,60),(30,57),(31,58),(32,59),(33,54),(34,55),(35,56),(36,53)], [(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,17,61,46),(2,42,62,10),(3,19,63,48),(4,44,64,12),(5,49,27,22),(6,13,28,37),(7,51,25,24),(8,15,26,39),(9,58,41,29),(11,60,43,31),(14,35,38,54),(16,33,40,56),(18,30,47,59),(20,32,45,57),(21,55,52,36),(23,53,50,34)], [(1,51,58,14),(2,21,59,39),(3,49,60,16),(4,23,57,37),(5,41,56,17),(6,10,53,47),(7,43,54,19),(8,12,55,45),(9,33,46,27),(11,35,48,25),(13,64,50,32),(15,62,52,30),(18,28,42,34),(20,26,44,36),(22,31,40,63),(24,29,38,61)], [(1,34),(2,7),(3,36),(4,5),(6,29),(8,31),(9,50),(10,38),(11,52),(12,40),(13,46),(14,42),(15,48),(16,44),(17,37),(18,51),(19,39),(20,49),(21,43),(22,45),(23,41),(24,47),(25,62),(26,60),(27,64),(28,58),(30,35),(32,33),(53,61),(54,59),(55,63),(56,57)]])

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₄₃C₂⁴

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₄₃C₂⁴ ►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	1	0
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	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	4	0
0	0	0	0	0	4

0	3	0	0	0	0
2	0	0	0	0	0
0	0	2	0	0	0
0	0	3	3	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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

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

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

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

C₂³.₆₄₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{643}C_2^4

% in TeX

G:=Group("C2^3.643C2^4");

// GroupNames label

G:=SmallGroup(128,1475);

// by ID

G=gap.SmallGroup(128,1475);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.643C24 order 128 = 27

360th central stem extension by C23 of C24

G = C₂³.₆₄₃C₂⁴ order 128 = 2⁷

360^th central stem extension by C₂³ of C₂⁴