C2^3.652C2^4 - GroupNames

G = C₂³.₆₅₂C₂⁴ order 128 = 2⁷

369^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₀C₂³, C₂³.₆₅₂C₂⁴, C₂².₄₂₅2₊¹⁺⁴, C₂³⋊₂D₄.₂₉C₂, C₂³.₄Q₈⋊₆₀C₂, C₂³.₁₈₉(C₄○D₄), C₂³.₃₄D₄⋊₅₈C₂, (C₂³×C₄).₁₆₃C₂², (C₂²×C₄).₅₇₃C₂³, (C₂×C₄²).₆₉₁C₂², C₂³.₁₀D₄⋊₁₀₀C₂, C₂³.₂₃D₄⋊₁₀₂C₂, C₂³.₁₁D₄⋊₁₁₁C₂, C₂⁴.₃C₂²⋊₉₂C₂, (C₂²×D₄).₂₆₉C₂², C₂⁴.C₂²⋊₁₆₀C₂, C₂.₈₅(C₂².₃₂C₂⁴), C₂³.₆₃C₂³⋊₁₆₇C₂, C₂.₂₇(C₂².₅₄C₂⁴), C₂.C₄².₃₅₆C₂², C₂.₁₀₄(C₂².₄₅C₂⁴), C₂.₃₈(C₂².₅₃C₂⁴), C₂.₅₄(C₂².₃₄C₂⁴), (C₂×C₄).₂₁₆(C₄○D₄), (C₂×C₄⋊C₄).₄₆₃C₂², C₂².₅₁₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₆₇C₂², SmallGroup(128,1484)

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

Derived series

C₁ — C₂³ — C₂³.₆₅₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂³.₆₅₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₅₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₅₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₅₂C₂⁴

Generators and relations for C₂³.₆₅₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=ca=ac, e²=a, f²=ba=ab, ede^-1=ad=da, geg=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=abd, fg=gf >

Subgroups: 548 in 242 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, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₃₄D₄, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₄Q₈, C₂³.₆₅₂C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₅₃C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₅₂C₂⁴

Smallest permutation representation of C₂³.₆₅₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₆₅₂C₂⁴

C₂³.₃₄D₄

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₄Q₈

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₅₂C₂⁴ ►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	1	0	0	0
0	0	0	1	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	1	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	0	2
0	0	0	0	3	0

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

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

4	0	0	0	0	0
4	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

C₂³.₆₅₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{652}C_2^4

% in TeX

G:=Group("C2^3.652C2^4");

// GroupNames label

G:=SmallGroup(128,1484);

// by ID

G=gap.SmallGroup(128,1484);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,268,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.652C24 order 128 = 27

369th central stem extension by C23 of C24

G = C₂³.₆₅₂C₂⁴ order 128 = 2⁷

369^th central stem extension by C₂³ of C₂⁴