C2^3.571C2^4 - GroupNames

G = C₂³.₅₇₁C₂⁴ order 128 = 2⁷

288^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₅₆C₂³, C₂³.₅₇₁C₂⁴, C₂².₃₄₅2₊¹⁺⁴, C₂².₂₅₇2_-¹⁺⁴, C₂.₃₅D₄², (C₂×D₄)⋊₁₃D₄, C₂²⋊C₄⋊₁₀D₄, C₂³.₅₈(C₂×D₄), C₂³⋊₂D₄⋊₃₄C₂, C₂.₅₄(D₄⋊₆D₄), C₂.₈₃(D₄⋊₅D₄), C₂³.₄Q₈⋊₃₉C₂, C₂³.₈Q₈⋊₉₂C₂, C₂³.₇Q₈⋊₈₁C₂, C₂³.₁₀D₄⋊₇₁C₂, C₂³.₂₃D₄⋊₇₇C₂, C₂.₃₇(C₂³⋊₃D₄), (C₂³×C₄).₄₄₁C₂², (C₂²×C₄).₈₆₀C₂³, C₂².₃₈₀(C₂²×D₄), (C₂²×D₄).₂₁₂C₂², C₂³.₈₁C₂³⋊₇₅C₂, C₂.C₄².₂₈₂C₂², C₂.₆(C₂².₅₆C₂⁴), C₂.₃₈(C₂².₃₄C₂⁴), C₂.₃₆(C₂².₃₁C₂⁴), (C₂×C₄⋊D₄)⋊₂₉C₂, (C₂×C₄).₄₁₁(C₂×D₄), (C₂×C₄).₄₁₅(C₄○D₄), (C₂×C₄⋊C₄).₃₈₉C₂², C₂².₄₃₇(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₂₈C₂, (C₂×C₂²⋊C₄).₂₄₂C₂², SmallGroup(128,1403)

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

Derived series

C₁ — C₂³ — C₂³.₅₇₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂².D₄ — C₂³.₅₇₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₇₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₇₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₇₁C₂⁴

Generators and relations for C₂³.₅₇₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=g²=1, f²=a, ab=ba, ac=ca, ede=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: 756 in 341 conjugacy classes, 104 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂³.₅₇₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₁C₂⁴, C₂².₃₄C₂⁴, D₄², D₄⋊₅D₄, D₄⋊₆D₄, C₂².₅₆C₂⁴, C₂³.₅₇₁C₂⁴

Smallest permutation representation of C₂³.₅₇₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	···	2M	2N	4A	···	4L	4M	···	4Q
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

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₅₇₁C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂²⋊C₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₅₇₁C₂⁴ ►in GL₆(𝔽₅)

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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

C₂³.₅₇₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{571}C_2^4

% in TeX

G:=Group("C2^3.571C2^4");

// GroupNames label

G:=SmallGroup(128,1403);

// by ID

G=gap.SmallGroup(128,1403);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=g^2=1,f^2=a,a*b=b*a,a*c=c*a,e*d*e=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 = C23.571C24 order 128 = 27

288th central stem extension by C23 of C24

G = C₂³.₅₇₁C₂⁴ order 128 = 2⁷

288^th central stem extension by C₂³ of C₂⁴