C2^3.681C2^4 - GroupNames

G = C₂³.₆₈₁C₂⁴ order 128 = 2⁷

398^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₉C₂³, C₂³.₆₈₁C₂⁴, C₂².₃₄₅2_-¹⁺⁴, C₂².₄₅₄2₊¹⁺⁴, C₄²⋊₈C₄⋊₆₈C₂, C₂³.₉₈(C₄○D₄), C₂³.Q₈⋊₈₅C₂, (C₂³×C₄).₄₉₃C₂², (C₂²×C₄).₅₉₅C₂³, (C₂×C₄²).₁₀₇C₂², C₂³.₇Q₈⋊₁₁₀C₂, C₂³.₈Q₈⋊₁₃₅C₂, C₂³.₁₁D₄⋊₁₁₈C₂, C₂³.₁₀D₄.₆₂C₂, C₂³.₂₃D₄.₇₂C₂, (C₂²×D₄).₂₇₈C₂², C₂⁴.C₂²⋊₁₆₇C₂, C₂³.₈₃C₂³⋊₁₁₇C₂, C₂³.₆₃C₂³⋊₁₈₂C₂, C₂³.₈₁C₂³⋊₁₂₄C₂, C₂.₁₀₁(C₂².₃₂C₂⁴), C₂.C₄².₃₈₅C₂², C₂.₄₃(C₂².₄₉C₂⁴), C₂.₃₉(C₂².₅₆C₂⁴), C₂.₁₁₇(C₂².₄₆C₂⁴), C₂.₁₁₀(C₂².₄₇C₂⁴), C₂.₁₀₁(C₂².₃₃C₂⁴), (C₂×C₄).₄₆₈(C₄○D₄), (C₂×C₄⋊C₄).₄₉₁C₂², C₂².₅₄₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₇C₂², SmallGroup(128,1513)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₈₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₁C₂⁴

Generators and relations for C₂³.₆₈₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=cb=bc, f²=a, g²=ba=ab, ac=ca, ede=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 452 in 217 conjugacy classes, 88 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, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₈₁C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₈₁C₂⁴

Smallest permutation representation of C₂³.₆₈₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₈₁C₂⁴

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₈₁C₂⁴ ►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	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

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

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

0	3	0	0	0	0
2	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	1	0

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

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

C₂³.₆₈₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{681}C_2^4

% in TeX

G:=Group("C2^3.681C2^4");

// GroupNames label

G:=SmallGroup(128,1513);

// by ID

G=gap.SmallGroup(128,1513);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,758,723,100,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.681C24 order 128 = 27

398th central stem extension by C23 of C24

G = C₂³.₆₈₁C₂⁴ order 128 = 2⁷

398^th central stem extension by C₂³ of C₂⁴