C2^3.640C2^4 - GroupNames

G = C₂³.₆₄₀C₂⁴ order 128 = 2⁷

357^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₇₇C₂³, C₂³.₆₄₀C₂⁴, C₂².₄₁₃2₊¹⁺⁴, C₂².₃₁₂2_-¹⁺⁴, C₂³.Q₈⋊₇₆C₂, C₂³.₄Q₈⋊₅₅C₂, C₂³.₁₈₄(C₄○D₄), C₂³.₃₄D₄⋊₅₂C₂, (C₂×C₄²).₆₈₇C₂², (C₂³×C₄).₄₈₂C₂², (C₂²×C₄).₂₀₂C₂³, C₂³.₇Q₈⋊₁₀₀C₂, C₂³.₁₁D₄⋊₁₀₅C₂, C₂³.₂₃D₄.₆₁C₂, C₂³.₁₀D₄.₅₃C₂, (C₂²×D₄).₂₆₁C₂², C₂⁴.C₂²⋊₁₅₃C₂, C₂⁴.₃C₂².₆₈C₂, C₂³.₆₃C₂³⋊₁₅₆C₂, C₂³.₈₁C₂³⋊₁₀₆C₂, C₂.₉₂(C₂².₄₅C₂⁴), C₂.C₄².₃₄₄C₂², C₂.₄₉(C₂².₃₄C₂⁴), C₂.₅₉(C₂².₅₀C₂⁴), C₂.₂₆(C₂².₅₆C₂⁴), C₂.₈₄(C₂².₃₃C₂⁴), (C₂×C₄).₂₁₄(C₄○D₄), (C₂×C₄⋊C₄).₄₅₁C₂², C₂².₅₀₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₀C₂², SmallGroup(128,1472)

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

Derived series

C₁ — C₂³ — C₂³.₆₄₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂³.₆₄₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₄₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₄₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₄₀C₂⁴

Generators and relations for C₂³.₆₄₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=ca=ac, e²=ba=ab, f²=a, ede^-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, 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: 452 in 217 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, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₆₄₀C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₄₀C₂⁴

Smallest permutation representation of C₂³.₆₄₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4P	4Q	···	4U
order	1	2	···	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₄₀C₂⁴

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₄₀C₂⁴ ►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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	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	0	0	0	1

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

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

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

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

C₂³.₆₄₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{640}C_2^4

% in TeX

G:=Group("C2^3.640C2^4");

// GroupNames label

G:=SmallGroup(128,1472);

// by ID

G=gap.SmallGroup(128,1472);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,100,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c*a=a*c,e^2=b*a=a*b,f^2=a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,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 = C23.640C24 order 128 = 27

357th central stem extension by C23 of C24

G = C₂³.₆₄₀C₂⁴ order 128 = 2⁷

357^th central stem extension by C₂³ of C₂⁴