C2^3.671C2^4 - GroupNames

G = C₂³.₆₇₁C₂⁴ order 128 = 2⁷

388^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₇₁C₂⁴, C₂⁴.₄₄₆C₂³, C₂².₃₃₇2_-¹⁺⁴, C₂².₄₄₄2₊¹⁺⁴, C₂³.₁₉₄(C₄○D₄), (C₂²×C₄).₅₈₈C₂³, (C₂³×C₄).₁₇₁C₂², (C₂×C₄²).₇₀₂C₂², C₂³.₈Q₈.₆₃C₂, C₂³.₄Q₈.₂₆C₂, C₂³.₃₄D₄.₃₄C₂, C₂³.₁₁D₄.₅₀C₂, C₂.₉₅(C₂².₃₂C₂⁴), C₂⁴.C₂².₇₀C₂, C₂³.₆₃C₂³⋊₁₇₅C₂, C₂³.₈₃C₂³⋊₁₀₉C₂, C₂³.₆₅C₂³⋊₁₄₆C₂, C₂.C₄².₃₇₅C₂², C₂.₄₃(C₂².₅₇C₂⁴), C₂.₄₀(C₂².₅₃C₂⁴), C₂.₉₆(C₂².₃₃C₂⁴), C₂.₁₀₉(C₂².₄₆C₂⁴), (C₂×C₄).₂₂₁(C₄○D₄), (C₂×C₄⋊C₄).₄₈₁C₂², C₂².₅₃₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₂C₂², SmallGroup(128,1503)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₈Q₈ — C₂³.₆₇₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₁C₂⁴

Generators and relations for C₂³.₆₇₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=dc=cd, f²=g²=cb=bc, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 372 in 196 conjugacy classes, 88 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, C₂×C₄⋊C₄, C₂³×C₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₇₁C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₄₆C₂⁴, C₂².₅₃C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₇₁C₂⁴

Smallest permutation representation of C₂³.₆₇₁C₂⁴
►On 64 points

Generators in S₆₄

(2 38)(4 40)(6 24)(8 22)(10 28)(12 26)(13 45)(14 60)(15 47)(16 58)(17 49)(18 64)(19 51)(20 62)(29 57)(30 48)(31 59)(32 46)(33 61)(34 52)(35 63)(36 50)(42 56)(44 54)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₇₁C₂⁴

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₇₁C₂⁴ ►in GL₆(𝔽₅)

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	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

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

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

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

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

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

C₂³.₆₇₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{671}C_2^4

% in TeX

G:=Group("C2^3.671C2^4");

// GroupNames label

G:=SmallGroup(128,1503);

// by ID

G=gap.SmallGroup(128,1503);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,344,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.671C24 order 128 = 27

388th central stem extension by C23 of C24

G = C₂³.₆₇₁C₂⁴ order 128 = 2⁷

388^th central stem extension by C₂³ of C₂⁴