C2^3.659C2^4 - GroupNames

G = C₂³.₆₅₉C₂⁴ order 128 = 2⁷

376^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₃C₂³, C₂³.₆₅₉C₂⁴, C₂².₄₃₂2₊¹⁺⁴, C₂².₃₂₆2_-¹⁺⁴, C₄²⋊₅C₄⋊₃₂C₂, C₂³⋊Q₈.₂₅C₂, (C₂×C₄²).₁₀₁C₂², (C₂²×C₄).₅₇₉C₂³, C₂³.₁₁D₄.₄₆C₂, (C₂²×Q₈).₂₁₁C₂², C₂³.₈₄C₂³⋊₁₁C₂, C₂³.₆₇C₂³⋊₉₈C₂, C₂³.₇₈C₂³⋊₅₆C₂, C₂.₈₈(C₂².₃₂C₂⁴), C₂⁴.C₂².₆₄C₂, C₂³.₆₃C₂³⋊₁₇₀C₂, C₂.C₄².₃₆₃C₂², C₂.₁₁₁(C₂².₄₅C₂⁴), C₂.₆₂(C₂².₅₀C₂⁴), C₂.₃₆(C₂².₅₇C₂⁴), C₂.₁₀₁(C₂².₃₆C₂⁴), (C₂×C₄).₂₁₉(C₄○D₄), (C₂×C₄⋊C₄).₄₇₀C₂², C₂².₅₂₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₈C₂², SmallGroup(128,1491)

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

Derived series

C₁ — C₂³ — C₂³.₆₅₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₇C₂³ — C₂³.₆₅₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₅₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₅₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₅₉C₂⁴

Generators and relations for C₂³.₆₅₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=g²=a, f²=ba=ab, ede^-1=ad=da, geg^-1=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^-1=abd, fg=gf >

Subgroups: 388 in 197 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×Q₈, C₄²⋊₅C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂³.₈₄C₂³, C₂³.₆₅₉C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₅₉C₂⁴

Smallest permutation representation of C₂³.₆₅₉C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4R	4S	···	4W
order	1	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₅₉C₂⁴

C₄²⋊₅C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂³.₈₄C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₅₉C₂⁴ ►in GL₆(𝔽₅)

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

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

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	2	0	0
0	0	2	0	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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	3	0
0	0	0	0	1	2

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

C₂³.₆₅₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{659}C_2^4

% in TeX

G:=Group("C2^3.659C2^4");

// GroupNames label

G:=SmallGroup(128,1491);

// by ID

G=gap.SmallGroup(128,1491);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.659C24 order 128 = 27

376th central stem extension by C23 of C24

G = C₂³.₆₅₉C₂⁴ order 128 = 2⁷

376^th central stem extension by C₂³ of C₂⁴