C2^3.431C2^4 - GroupNames

G = C₂³.₄₃₁C₂⁴ order 128 = 2⁷

148^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₂₄C₂³, C₂³.₄₃₁C₂⁴, C₂².₂₂₂2₊¹⁺⁴, C₄²⋊₄C₄⋊₂₄C₂, C₂³.₄₈(C₄○D₄), (C₂×C₄²).₆₀C₂², (C₂²×C₄).₉₁C₂³, C₂³.₇Q₈⋊₆₁C₂, C₂³.₁₁D₄⋊₃₉C₂, (C₂³×C₄).₃₈₈C₂², C₂⁴.C₂²⋊₇₅C₂, C₂³.₂₃D₄.₃₃C₂, C₂³.₁₀D₄.₁₇C₂, (C₂²×D₄).₁₅₉C₂², C₂³.₈₁C₂³⋊₃₃C₂, C₂³.₆₃C₂³⋊₈₂C₂, C₂⁴.₃C₂².₄₂C₂, C₂.₄₃(C₂².₄₅C₂⁴), C₂.C₄².₅₄₄C₂², C₂.₅₁(C₂².₄₇C₂⁴), C₂.₇₄(C₂³.₃₆C₂³), C₂.₂₂(C₂².₄₉C₂⁴), C₂.₂₃(C₂².₃₄C₂⁴), (C₄×C₂²⋊C₄)⋊₈₂C₂, (C₂×C₄).₃₈₃(C₄○D₄), (C₂×C₄⋊C₄).₂₉₃C₂², C₂².₃₀₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₇₀C₂², SmallGroup(128,1263)

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

Derived series

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂³.₄₃₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₃₁C₂⁴

Generators and relations for C₂³.₄₃₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=ca=ac, e²=b, g²=ba=ab, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, 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, C₂³.₇Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄₃₁C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂³.₄₃₁C₂⁴

Smallest permutation representation of C₂³.₄₃₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

C₂³.₄₃₁C₂⁴

C₄²⋊₄C₄

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₃₁C₂⁴ ►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	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	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	3	0	0	0	0
3	0	0	0	0	0
0	0	0	4	0	0
0	0	1	0	0	0
0	0	0	0	3	0
0	0	0	0	0	2

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

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

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

C₂³.₄₃₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{431}C_2^4

% in TeX

G:=Group("C2^3.431C2^4");

// GroupNames label

G:=SmallGroup(128,1263);

// by ID

G=gap.SmallGroup(128,1263);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,675,192]);

// Polycyclic

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

// generators/relations

G = C23.431C24 order 128 = 27

148th central stem extension by C23 of C24

G = C₂³.₄₃₁C₂⁴ order 128 = 2⁷

148^th central stem extension by C₂³ of C₂⁴