C2^3.683C2^4 - GroupNames

G = C₂³.₆₈₃C₂⁴ order 128 = 2⁷

400^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₉₀C₂³, C₂³.₆₈₃C₂⁴, C₂².₃₄₇2_-¹⁺⁴, C₂².₄₅₆2₊¹⁺⁴, C₄²⋊₈C₄⋊₇₀C₂, (C₂²×C₄).₅₉₇C₂³, (C₂×C₄²).₁₀₈C₂², C₂³.Q₈.₄₀C₂, C₂³.₁₁D₄.₅₅C₂, C₂⁴.C₂².₇₅C₂, C₂³.₈₁C₂³⋊₁₂₅C₂, C₂³.₆₅C₂³⋊₁₅₃C₂, C₂³.₆₃C₂³⋊₁₈₃C₂, C₂³.₈₃C₂³⋊₁₁₉C₂, C₂.C₄².₃₈₇C₂², C₂.₄₄(C₂².₄₉C₂⁴), C₂.₄₀(C₂².₅₆C₂⁴), C₂.₅₉(C₂².₃₅C₂⁴), C₂.₆₈(C₂².₅₀C₂⁴), C₂.₁₀₃(C₂².₃₃C₂⁴), C₂.₁₁₂(C₂².₄₇C₂⁴), (C₂×C₄).₂₂₈(C₄○D₄), (C₂×C₄⋊C₄).₄₉₃C₂², C₂².₅₄₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₉C₂², SmallGroup(128,1515)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₃C₂³ — C₂³.₆₈₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₃C₂⁴

Generators and relations for C₂³.₆₈₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=abc, e²=f²=a, g²=ba=ab, ac=ca, ede^-1=ad=da, geg^-1=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, gdg^-1=abd, fg=gf >

Subgroups: 356 in 189 conjugacy classes, 88 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⋊₈C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₆₈₃C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₈₃C₂⁴

Smallest permutation representation of C₂³.₆₈₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4R	4S	···	4W
order	1	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₈₃C₂⁴

C₄²⋊₈C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₈₃C₂⁴ ►in GL₆(𝔽₅)

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

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	1	0
0	0	0	0	0	1

0	2	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	4
0	0	0	0	1	0

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

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

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

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

C₂³.₆₈₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{683}C_2^4

% in TeX

G:=Group("C2^3.683C2^4");

// GroupNames label

G:=SmallGroup(128,1515);

// by ID

G=gap.SmallGroup(128,1515);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,100,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.683C24 order 128 = 27

400th central stem extension by C23 of C24

G = C₂³.₆₈₃C₂⁴ order 128 = 2⁷

400^th central stem extension by C₂³ of C₂⁴