C2^3.429C2^4 - GroupNames

G = C₂³.₄₂₉C₂⁴ order 128 = 2⁷

146^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₂₃C₂³, C₂³.₄₂₉C₂⁴, C₂².₂₂₀2₊¹⁺⁴, C₂².₁₆₉2_-¹⁺⁴, C₄²⋊₄C₄⋊₂₂C₂, C₄²⋊₈C₄⋊₃₉C₂, (C₂×C₄²).₅₈C₂², C₂³.Q₈.₁₂C₂, (C₂²×C₄).₁₂₅₈C₂³, C₂³.₁₁D₄.₁₆C₂, C₂³.₈₁C₂³⋊₃₂C₂, C₂³.₈₃C₂³⋊₃₄C₂, C₂³.₆₅C₂³⋊₈₂C₂, C₂³.₆₃C₂³⋊₈₀C₂, C₂⁴.C₂².₂₈C₂, C₂.C₄².₅₄₃C₂², C₂.₅₆(C₂².₄₆C₂⁴), C₂.₂₁(C₂².₄₉C₂⁴), C₂.₄₁(C₂².₃₆C₂⁴), C₂.₄₉(C₂².₄₇C₂⁴), C₂.₇₂(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₈₃C₂, (C₂×C₄).₃₈₁(C₄○D₄), (C₂×C₄⋊C₄).₂₉₁C₂², C₂².₃₀₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₆₈C₂², SmallGroup(128,1261)

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

Derived series

C₁ — C₂³ — C₂³.₄₂₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₂₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₂₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₂₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₂₉C₂⁴

Generators and relations for C₂³.₄₂₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=ba=ab, f²=a, g²=b, ede^-1=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 356 in 199 conjugacy classes, 92 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⋊₄C₄, C₄×C₄⋊C₄, C₄²⋊₈C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₄₂₉C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂³.₄₂₉C₂⁴

Smallest permutation representation of C₂³.₄₂₉C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4H	4I	···	4Z	4AA	4AB	4AC
order	1	2	···	2	2	4	···	4	4	···	4	4	4	4
size	1	1	···	1	8	2	···	2	4	···	4	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₄₂₉C₂⁴

C₄²⋊₄C₄

C₄×C₄⋊C₄

C₄²⋊₈C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₂₉C₂⁴ ►in GL₆(𝔽₅)

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	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

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

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

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

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

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

C₂³.₄₂₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{429}C_2^4

% in TeX

G:=Group("C2^3.429C2^4");

// GroupNames label

G:=SmallGroup(128,1261);

// by ID

G=gap.SmallGroup(128,1261);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,100,675,136]);

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

// generators/relations

G = C23.429C24 order 128 = 27

146th central stem extension by C23 of C24

G = C₂³.₄₂₉C₂⁴ order 128 = 2⁷

146^th central stem extension by C₂³ of C₂⁴