C2^3.641C2^4 - GroupNames

G = C₂³.₆₄₁C₂⁴ order 128 = 2⁷

358^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₇₈C₂³, C₂³.₆₄₁C₂⁴, C₂².₃₁₃2_-¹⁺⁴, C₂².₄₁₄2₊¹⁺⁴, C₂³.₈₇(C₄○D₄), (C₂×C₄²).₉₀C₂², C₂³.₄Q₈⋊₅₆C₂, (C₂³×C₄).₄₈₃C₂², (C₂²×C₄).₅₆₆C₂³, C₂³.₇Q₈⋊₁₀₁C₂, C₂³.₁₁D₄⋊₁₀₆C₂, C₂³.₂₃D₄.₆₂C₂, C₂³.₁₀D₄.₅₄C₂, (C₂²×D₄).₂₆₂C₂², C₂⁴.C₂²⋊₁₅₄C₂, C₂³.₈₃C₂³⋊₉₃C₂, C₂.₇₉(C₂².₃₂C₂⁴), C₂³.₈₁C₂³⋊₁₀₇C₂, C₂³.₆₃C₂³⋊₁₅₇C₂, C₂.₉₃(C₂².₄₅C₂⁴), C₂.C₄².₃₄₅C₂², C₂.₉₂(C₂².₄₇C₂⁴), C₂.₉₁(C₂².₄₆C₂⁴), C₂.₉₁(C₂².₃₆C₂⁴), C₂.₅₀(C₂².₃₄C₂⁴), C₂.₂₇(C₂².₅₆C₂⁴), (C₂×C₄).₄₄₂(C₄○D₄), (C₂×C₄⋊C₄).₄₅₂C₂², C₂².₅₀₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₁C₂², SmallGroup(128,1473)

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

Derived series

C₁ — C₂³ — C₂³.₆₄₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂³.₆₄₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₄₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₄₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₄₁C₂⁴

Generators and relations for C₂³.₆₄₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=ca=ac, e²=b, g²=a, ab=ba, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=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, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₄₁C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₄₁C₂⁴

Smallest permutation representation of C₂³.₆₄₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

C₂³.₆₄₁C₂⁴

C₂³.₇Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₄₁C₂⁴ ►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	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

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

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

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

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

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

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

C₂³.₆₄₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{641}C_2^4

% in TeX

G:=Group("C2^3.641C2^4");

// GroupNames label

G:=SmallGroup(128,1473);

// by ID

G=gap.SmallGroup(128,1473);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.641C24 order 128 = 27

358th central stem extension by C23 of C24

G = C₂³.₆₄₁C₂⁴ order 128 = 2⁷

358^th central stem extension by C₂³ of C₂⁴