C2^3.500C2^4 - GroupNames

G = C₂³.₅₀₀C₂⁴ order 128 = 2⁷

217^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₅₀₀C₂⁴, C₂⁴.₃₅₁C₂³, C₂².₂₈₁2₊¹⁺⁴, (C₂²×C₄)⋊₁₃D₄, C₂³.₁₈₅(C₂×D₄), C₂³.₆₁(C₄○D₄), C₂³⋊₂D₄.₁₃C₂, C₂³.₈Q₈⋊₇₇C₂, C₂³.₂₃D₄⋊₆₂C₂, C₂³.₃₄D₄⋊₃₉C₂, C₂³.₁₁D₄⋊₅₂C₂, C₂³.₁₀D₄⋊₄₈C₂, C₂.₁₉(C₂³⋊₃D₄), (C₂²×C₄).₁₂₂C₂³, (C₂³×C₄).₁₃₁C₂², (C₂×C₄²).₅₈₇C₂², C₂².₃₃₀(C₂²×D₄), C₂⁴.C₂²⋊₉₈C₂, (C₂²×D₄).₅₃₇C₂², C₂.₇₃(C₂².₁₉C₂⁴), C₂³.₆₃C₂³⋊₁₀₄C₂, C₂.₆₆(C₂².₄₅C₂⁴), C₂.C₄².₂₃₀C₂², C₂.₇₆(C₂².₄₇C₂⁴), C₂.₃₀(C₂².₅₃C₂⁴), (C₂×C₄×D₄)⋊₄₉C₂, (C₂×C₄).₁₁₉₈(C₂×D₄), (C₂×C₄).₁₆₀(C₄○D₄), (C₂×C₄⋊C₄).₃₄₀C₂², C₂².₃₇₆(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₀₁C₂², SmallGroup(128,1332)

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

Derived series

C₁ — C₂³ — C₂³.₅₀₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂³.₅₀₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₀₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₀₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₀₀C₂⁴

Generators and relations for C₂³.₅₀₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=db=bd, eae=ab=ba, ac=ca, faf=ad=da, ag=ga, bc=cb, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 612 in 302 conjugacy classes, 100 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, C₄×D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂×C₄×D₄, C₂³.₅₀₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₁₉C₂⁴, C₂³⋊₃D₄, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₃C₂⁴, C₂³.₅₀₀C₂⁴

Smallest permutation representation of C₂³.₅₀₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

(5 12)(6 9)(7 10)(8 11)(13 61)(14 62)(15 63)(16 64)(29 37)(30 38)(31 39)(32 40)(33 42)(34 43)(35 44)(36 41)(45 60)(46 57)(47 58)(48 59)(49 54)(50 55)(51 56)(52 53)

(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)

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	···	2M	4A	···	4H	4I	···	4T	4U	4V	4W	4X
order	1	2	···	2	2	···	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	···	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₅₀₀C₂⁴

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂×C₄×D₄

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₀₀C₂⁴ ►in GL₆(𝔽₅)

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

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	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

C₂³.₅₀₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{500}C_2^4

% in TeX

G:=Group("C2^3.500C2^4");

// GroupNames label

G:=SmallGroup(128,1332);

// by ID

G=gap.SmallGroup(128,1332);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,675,304]);

// Polycyclic

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

// generators/relations

G = C23.500C24 order 128 = 27

217th central stem extension by C23 of C24

G = C₂³.₅₀₀C₂⁴ order 128 = 2⁷

217^th central stem extension by C₂³ of C₂⁴