C2^4.584C2^3 - GroupNames

G = C₂⁴.₅₈₄C₂³ order 128 = 2⁷

65^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂²=C₂

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

Aliases: C₂⁴.₅₈₄C₂³, C₂³.₄₆₉C₂⁴, C₂².₂₅₂2₊¹⁺⁴, C₂³.₉₉(C₂×Q₈), (C₂²×C₄).₅₉Q₈, (C₂×C₄²).₆₆C₂², C₂.₅(C₂³⋊₂Q₈), C₂³.₃₁₅(C₄○D₄), (C₂³×C₄).₄₀₆C₂², (C₂²×C₄).₅₄₄C₂³, C₂³.₈Q₈.₃₁C₂, C₂³.₇Q₈.₅₃C₂, C₂².₁₁₀(C₂²×Q₈), C₂².₁₅(C₄².C₂), C₂³.₈₃C₂³⋊₄₁C₂, C₂³.₆₃C₂³⋊₈₉C₂, C₂³.₈₁C₂³⋊₄₂C₂, C₂.₅₂(C₂².₄₅C₂⁴), C₂.C₄².₂₀₅C₂², C₂.₂₈(C₂³.₃₇C₂³), C₂.₅₉(C₂².₄₇C₂⁴), (C₂×C₄).₁₆₅(C₂×Q₈), (C₄×C₂²⋊C₄).₆₃C₂, C₂.₁₆(C₂×C₄².C₂), (C₂×C₄).₃₉₃(C₄○D₄), (C₂×C₄⋊C₄).₃₁₆C₂², C₂².₃₄₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₆₉C₂², (C₂×C₂.C₄²).₂₉C₂, SmallGroup(128,1301)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₈₄C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂.C₄² — C₂⁴.₅₈₄C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₈₄C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₈₄C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₈₄C₂³

Generators and relations for C₂⁴.₅₈₄C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=cb=bc, e²=ca=ac, f²=b, ab=ba, ede^-1=gdg=ad=da, 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, eg=ge, fg=gf >

Subgroups: 420 in 228 conjugacy classes, 104 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³×C₄, C₂×C₂.C₄², C₄×C₂²⋊C₄, C₂³.₇Q₈, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₅₈₄C₂³
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₄².C₂, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₄².C₂, C₂³.₃₇C₂³, C₂³⋊₂Q₈, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₅₈₄C₂³

Smallest permutation representation of C₂⁴.₅₈₄C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4V	4W	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

Q₈

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₅₈₄C₂³

C₂×C₂.C₄²

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₅₈₄C₂³ ►in GL₆(𝔽₅)

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	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	4	0
0	0	0	0	0	4

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	1	0	0	0	0
1	0	0	0	0	0
0	0	0	1	0	0
0	0	4	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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	0	1
0	0	0	0	4	0

1	0	0	0	0	0
0	4	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

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

C₂⁴.₅₈₄C₂³ in GAP, Magma, Sage, TeX

C_2^4._{584}C_2^3

% in TeX

G:=Group("C2^4.584C2^3");

// GroupNames label

G:=SmallGroup(128,1301);

// by ID

G=gap.SmallGroup(128,1301);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,184,675]);

// 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=c*a=a*c,f^2=b,a*b=b*a,e*d*e^-1=g*d*g=a*d=d*a,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,e*g=g*e,f*g=g*f>;

// generators/relations

G = C24.584C23 order 128 = 27

65th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₈₄C₂³ order 128 = 2⁷

65^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂²=C₂