C2^3.581C2^4 - GroupNames

G = C₂³.₅₈₁C₂⁴ order 128 = 2⁷

298^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₆₁C₂³, C₂³.₅₈₁C₂⁴, C₂².₂₆₄2_-¹⁺⁴, C₂².₃₅₅2₊¹⁺⁴, C₂.₄₅D₄², C₄⋊C₄⋊₁₀D₄, C₂²⋊C₄.₁₀D₄, C₂³⋊Q₈⋊₄₀C₂, C₂³.₂₀₇(C₂×D₄), C₂.₈₈(D₄⋊₅D₄), C₂.₄₄(Q₈⋊₅D₄), C₂³.₇₅(C₄○D₄), C₂³.₈Q₈⋊₉₈C₂, C₂³.Q₈⋊₅₂C₂, C₂³.₂₃D₄⋊₈₂C₂, C₂³.₁₀D₄⋊₇₇C₂, (C₂×C₄²).₆₃₈C₂², (C₂²×C₄).₈₆₈C₂³, (C₂³×C₄).₄₄₉C₂², C₂².₃₉₀(C₂²×D₄), C₂⁴.₃C₂²⋊₇₅C₂, (C₂²×D₄).₂₂₀C₂², (C₂²×Q₈).₁₇₇C₂², C₂⁴.C₂²⋊₁₂₁C₂, C₂³.₈₁C₂³⋊₇₈C₂, C₂³.₆₅C₂³⋊₁₁₅C₂, C₂.C₄².₂₉₀C₂², C₂.₉(C₂².₅₆C₂⁴), C₂.₄₀(C₂².₃₁C₂⁴), C₂.₅₈(C₂².₃₃C₂⁴), C₂.₅₅(C₂³.₃₈C₂³), (C₂×C₄).₈₉(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₃₆C₂, (C₂×C₄⋊D₄).₄₃C₂, (C₂×C₄⋊C₄).₃₉₇C₂², C₂².₄₄₄(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₂C₂, (C₂×C₂²⋊C₄).₂₅₀C₂², SmallGroup(128,1413)

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

Derived series

C₁ — C₂³ — C₂³.₅₈₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂².D₄ — C₂³.₅₈₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₈₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₈₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₈₁C₂⁴

Generators and relations for C₂³.₅₈₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=b, e²=f²=a, ab=ba, ac=ca, 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: 644 in 309 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₈₁C₂³, C₂×C₄⋊D₄, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂³.₅₈₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₈C₂³, C₂².₃₁C₂⁴, C₂².₃₃C₂⁴, D₄², D₄⋊₅D₄, Q₈⋊₅D₄, C₂².₅₆C₂⁴, C₂³.₅₈₁C₂⁴

Smallest permutation representation of C₂³.₅₈₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₅₈₁C₂⁴

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₈₁C₂³

C₂×C₄⋊D₄

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₂²⋊C₄

C₄⋊C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₈₁C₂⁴ ►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	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

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

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

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

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

C₂³.₅₈₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{581}C_2^4

% in TeX

G:=Group("C2^3.581C2^4");

// GroupNames label

G:=SmallGroup(128,1413);

// by ID

G=gap.SmallGroup(128,1413);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,100,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=b,e^2=f^2=a,a*b=b*a,a*c=c*a,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 = C23.581C24 order 128 = 27

298th central stem extension by C23 of C24

G = C₂³.₅₈₁C₂⁴ order 128 = 2⁷

298^th central stem extension by C₂³ of C₂⁴