C2^3.686C2^4 - GroupNames

G = C₂³.₆₈₆C₂⁴ order 128 = 2⁷

403^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₈₆C₂⁴, C₂⁴.₄₅₁C₂³, C₂².₃₄₉2_-¹⁺⁴, C₂².₄₅₉2₊¹⁺⁴, C₂³.Q₈⋊₈₇C₂, C₂³.₁₀₀(C₄○D₄), (C₂³×C₄).₄₉₅C₂², (C₂×C₄²).₇₁₃C₂², (C₂²×C₄).₅₉₉C₂³, C₂³.₈Q₈⋊₁₃₈C₂, C₂³.₇Q₈⋊₁₁₂C₂, C₂³.₁₁D₄⋊₁₂₀C₂, C₂³.₂₃D₄.₇₅C₂, C₂³.₁₀D₄.₆₅C₂, (C₂²×D₄).₂₈₂C₂², C₂⁴.C₂²⋊₁₇₀C₂, C₂⁴.₃C₂².₇₅C₂, C₂³.₈₁C₂³⋊₁₂₆C₂, C₂³.₆₃C₂³⋊₁₈₅C₂, C₂.₁₀₃(C₂².₃₂C₂⁴), C₂.C₄².₃₉₀C₂², C₂.₁₁₉(C₂².₄₅C₂⁴), C₂.₆₉(C₂².₅₀C₂⁴), C₂.₄₁(C₂².₅₆C₂⁴), C₂.₆₂(C₂².₃₄C₂⁴), C₂.₁₀₅(C₂².₃₃C₂⁴), C₂.₁₁₃(C₂².₄₇C₂⁴), (C₂×C₄).₂₂₉(C₄○D₄), (C₂×C₄⋊C₄).₄₉₆C₂², C₂².₅₄₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₂₂C₂², SmallGroup(128,1518)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₈₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₆C₂⁴

Generators and relations for C₂³.₆₈₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=abc, e²=b, g²=ba=ab, ac=ca, ede^-1=ad=da, geg^-1=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, 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, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₆₈₆C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₈₆C₂⁴

Smallest permutation representation of C₂³.₆₈₆C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

C₂³.₆₈₆C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₈₆C₂⁴ ►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	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	3	0
0	0	0	0	0	2

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	0	1
0	0	0	0	1	0

4	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	1	0
0	0	0	0	0	1

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

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

C₂³.₆₈₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{686}C_2^4

% in TeX

G:=Group("C2^3.686C2^4");

// GroupNames label

G:=SmallGroup(128,1518);

// by ID

G=gap.SmallGroup(128,1518);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,1571,346,192]);

// Polycyclic

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

// generators/relations

G = C23.686C24 order 128 = 27

403rd central stem extension by C23 of C24

G = C₂³.₆₈₆C₂⁴ order 128 = 2⁷

403^rd central stem extension by C₂³ of C₂⁴