C2^3.418C2^4 - GroupNames

G = C₂³.₄₁₈C₂⁴ order 128 = 2⁷

135^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₂₀C₂³, C₂³.₄₁₈C₂⁴, C₂².₂₁₂2₊¹⁺⁴, C₄²⋊₈C₄⋊₃₇C₂, C₄²⋊₅C₄⋊₁₄C₂, C₂³.₄₅(C₄○D₄), C₂³.Q₈⋊₂₄C₂, C₂³.₁₁D₄⋊₃₆C₂, (C₂×C₄²).₅₃₃C₂², (C₂²×C₄).₅₂₈C₂³, (C₂³×C₄).₁₀₇C₂², C₂⁴.C₂²⋊₇₂C₂, C₂³.₂₃D₄.₃₀C₂, C₂³.₁₀D₄.₁₄C₂, (C₂²×D₄).₁₅₆C₂², C₂³.₆₅C₂³⋊₇₇C₂, C₂³.₆₃C₂³⋊₇₂C₂, C₂.₂₁(C₂².₃₂C₂⁴), C₂.₃₅(C₂².₄₅C₂⁴), C₂.C₄².₁₆₆C₂², C₂.₁₉(C₂².₄₉C₂⁴), C₂.₆₁(C₂³.₃₆C₂³), C₂.₄₅(C₂².₄₇C₂⁴), (C₄×C₂²⋊C₄)⋊₇₉C₂, (C₂×C₄).₁₃₉(C₄○D₄), (C₂×C₄⋊C₄).₂₈₁C₂², C₂².₂₉₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₆₇C₂², SmallGroup(128,1250)

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

Derived series

C₁ — C₂³ — C₂³.₄₁₈C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₁₈C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₁₈C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₁₈C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₁₈C₂⁴

Generators and relations for C₂³.₄₁₈C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=ca=ac, f²=a, g²=b, ab=ba, ede=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: 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, C₄×C₂²⋊C₄, C₄²⋊₈C₄, C₄²⋊₅C₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₄₁₈C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂³.₄₁₈C₂⁴

Smallest permutation representation of C₂³.₄₁₈C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

C₂³.₄₁₈C₂⁴

C₄×C₂²⋊C₄

C₄²⋊₈C₄

C₄²⋊₅C₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₁₈C₂⁴ ►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	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

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

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

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

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

C₂³.₄₁₈C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{418}C_2^4

% in TeX

G:=Group("C2^3.418C2^4");

// GroupNames label

G:=SmallGroup(128,1250);

// by ID

G=gap.SmallGroup(128,1250);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c*a=a*c,f^2=a,g^2=b,a*b=b*a,e*d*e=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 = C23.418C24 order 128 = 27

135th central stem extension by C23 of C24

G = C₂³.₄₁₈C₂⁴ order 128 = 2⁷

135^th central stem extension by C₂³ of C₂⁴