C2^3.682C2^4 - GroupNames

G = C₂³.₆₈₂C₂⁴ order 128 = 2⁷

399^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₈₂C₂⁴, C₂⁴.₄₄₉C₂³, C₂².₄₅₅2₊¹⁺⁴, C₂².₃₄₆2_-¹⁺⁴, C₄²⋊₈C₄⋊₆₉C₂, C₂³.₉₉(C₄○D₄), (C₂³×C₄).₁₇₄C₂², (C₂×C₄²).₇₁₀C₂², (C₂²×C₄).₅₉₆C₂³, C₂³.₈Q₈⋊₁₃₆C₂, C₂³.₁₁D₄⋊₁₁₉C₂, C₂³.₂₃D₄.₇₃C₂, C₂³.₁₀D₄.₆₃C₂, (C₂²×D₄).₂₇₉C₂², C₂⁴.C₂²⋊₁₆₈C₂, C₂.₁₁(C₂⁴⋊C₂²), C₂³.₈₃C₂³⋊₁₁₈C₂, C₂.C₄².₃₈₆C₂², C₂.₁₁₈(C₂².₄₅C₂⁴), C₂.₁₀₂(C₂².₃₃C₂⁴), C₂.₁₁₁(C₂².₄₇C₂⁴), (C₂×C₄).₄₆₉(C₄○D₄), (C₂×C₄⋊C₄).₄₉₂C₂², C₂².₅₄₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₈C₂², SmallGroup(128,1514)

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

Derived series

C₁ — C₂³ — C₂³.₆₈₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₈₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₂C₂⁴

Generators and relations for C₂³.₆₈₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=bcd, f²=b, g²=cb=bc, eae^-1=gag^-1=ab=ba, ac=ca, faf^-1=ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=bce, fg=gf >

Subgroups: 468 in 224 conjugacy classes, 88 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂³.₈₃C₂³, C₂³.₆₈₂C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂⁴⋊C₂², C₂³.₆₈₂C₂⁴

Smallest permutation representation of C₂³.₆₈₂C₂⁴
►On 64 points

Generators in S₆₄

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

(1 60)(2 57)(3 58)(4 59)(5 30)(6 31)(7 32)(8 29)(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 41)(22 42)(23 43)(24 44)(25 36)(26 33)(27 34)(28 35)(53 61)(54 62)(55 63)(56 64)

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₈₂C₂⁴

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₈₂C₂⁴ ►in GL₆(𝔽₅)

0	1	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	3	4
0	0	0	0	3	2

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

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

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

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

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

C₂³.₆₈₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{682}C_2^4

% in TeX

G:=Group("C2^3.682C2^4");

// GroupNames label

G:=SmallGroup(128,1514);

// by ID

G=gap.SmallGroup(128,1514);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23.682C24 order 128 = 27

399th central stem extension by C23 of C24

G = C₂³.₆₈₂C₂⁴ order 128 = 2⁷

399^th central stem extension by C₂³ of C₂⁴