C2^3.426C2^4 - GroupNames

G = C₂³.₄₂₆C₂⁴ order 128 = 2⁷

143^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₂₆C₂⁴, C₂⁴.₃₁₄C₂³, C₂².₂₁₈2₊¹⁺⁴, C₄²⋊₄C₄⋊₂₀C₂, (C₂²×C₄).₈₈C₂³, (C₂×C₄²).₅₅C₂², C₂³.Q₈⋊₂₅C₂, C₂³.₁₄₈(C₄○D₄), C₂³.₁₁D₄⋊₃₈C₂, C₂³.₃₄D₄⋊₃₄C₂, (C₂³×C₄).₁₀₉C₂², C₂⁴.C₂²⋊₇₄C₂, C₂³.₂₃D₄.₃₂C₂, C₂³.₁₀D₄.₁₆C₂, (C₂²×D₄).₁₅₈C₂², C₂³.₆₃C₂³⋊₇₈C₂, C₂³.₈₃C₂³⋊₃₂C₂, C₂³.₆₅C₂³⋊₈₁C₂, C₂⁴.₃C₂².₄₁C₂, C₂.₄₀(C₂².₄₅C₂⁴), C₂.C₄².₅₄₂C₂², C₂.₂₀(C₂².₄₉C₂⁴), C₂.₆₉(C₂³.₃₆C₂³), C₂.₂₂(C₂².₃₄C₂⁴), C₂.₄₈(C₂².₄₇C₂⁴), (C₄×C₂²⋊C₄)⋊₈₁C₂, (C₂×C₄).₅₂₂(C₄○D₄), (C₂×C₄⋊C₄).₂₈₈C₂², C₂².₃₀₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₁C₂², SmallGroup(128,1258)

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

Derived series

C₁ — C₂³ — C₂³.₄₂₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₄₂₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₂₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₂₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₂₆C₂⁴

Generators and relations for C₂³.₄₂₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=cb=bc, g²=c, ab=ba, eae^-1=ac=ca, faf^-1=ad=da, ag=ga, bd=db, fef^-1=geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 452 in 227 conjugacy classes, 92 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, C₄²⋊₄C₄, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂³.₄₂₆C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂³.₄₂₆C₂⁴

Smallest permutation representation of C₂³.₄₂₆C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	4A	···	4H	4I	···	4X	4Y	4Z	4AA
order	1	2	···	2	2	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	4	4	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₂₆C₂⁴

C₄²⋊₄C₄

C₄×C₂²⋊C₄

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₂₆C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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	3	0
0	0	0	0	0	3

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

C₂³.₄₂₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{426}C_2^4

% in TeX

G:=Group("C2^3.426C2^4");

// GroupNames label

G:=SmallGroup(128,1258);

// by ID

G=gap.SmallGroup(128,1258);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,675,136]);

// Polycyclic

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

// generators/relations

G = C23.426C24 order 128 = 27

143rd central stem extension by C23 of C24

G = C₂³.₄₂₆C₂⁴ order 128 = 2⁷

143^rd central stem extension by C₂³ of C₂⁴