C2^3.672C2^4 - GroupNames

G = C₂³.₆₇₂C₂⁴ order 128 = 2⁷

389^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₄C₂³, C₂³.₆₇₂C₂⁴, C₂².₃₃₈2_-¹⁺⁴, C₂².₄₄₅2₊¹⁺⁴, C₄²⋊₈C₄⋊₆₅C₂, (C₂×C₄²).₇₀₃C₂², (C₂²×C₄).₅₈₉C₂³, C₂³.Q₈.₃₇C₂, C₂³.₁₁D₄.₅₁C₂, C₂⁴.C₂².₇₁C₂, C₂³.₈₁C₂³⋊₁₂₀C₂, C₂³.₆₅C₂³⋊₁₄₇C₂, C₂³.₆₃C₂³⋊₁₇₆C₂, C₂³.₈₃C₂³⋊₁₁₀C₂, C₂.C₄².₃₇₆C₂², C₂.₅₇(C₂².₃₅C₂⁴), C₂.₃₇(C₂².₅₆C₂⁴), C₂.₁₁₀(C₂².₄₆C₂⁴), C₂.₁₀₄(C₂².₄₇C₂⁴), C₂.₁₁₀(C₂².₃₆C₂⁴), (C₂×C₄).₂₂₂(C₄○D₄), (C₂×C₄⋊C₄).₄₈₂C₂², C₂².₅₃₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₃C₂², SmallGroup(128,1504)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₇₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₂C₂⁴

Generators and relations for C₂³.₆₇₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=abc, e²=f²=b, g²=a, ab=ba, ac=ca, 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: 356 in 189 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₈C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₇₂C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₅C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₇₂C₂⁴

Smallest permutation representation of C₂³.₆₇₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

C₂³.₆₇₂C₂⁴

C₄²⋊₈C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₇₂C₂⁴ ►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

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

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

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

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

C₂³.₆₇₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{672}C_2^4

% in TeX

G:=Group("C2^3.672C2^4");

// GroupNames label

G:=SmallGroup(128,1504);

// by ID

G=gap.SmallGroup(128,1504);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,184,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=f^2=b,g^2=a,a*b=b*a,a*c=c*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^-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 = C23.672C24 order 128 = 27

389th central stem extension by C23 of C24

G = C₂³.₆₇₂C₂⁴ order 128 = 2⁷

389^th central stem extension by C₂³ of C₂⁴