C2^3.400C2^4 - GroupNames

G = C₂³.₄₀₀C₂⁴ order 128 = 2⁷

117^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₁₃C₂³, C₂³.₄₀₀C₂⁴, C₂².₁₉₈2₊¹⁺⁴, C₄²⋊₈C₄⋊₃₀C₂, C₄²⋊₉C₄⋊₂₂C₂, C₂³.₁₈₄(C₂×D₄), (C₂²×C₄).₃₈₇D₄, (C₂×C₄²).₄₉C₂², C₂³.₁₀D₄⋊₃₇C₂, (C₂³×C₄).₃₈₄C₂², C₂².₂₇₆(C₂²×D₄), C₂⁴.C₂²⋊₆₇C₂, (C₂²×C₄).₁₄₈₃C₂³, C₂⁴.₃C₂²⋊₄₉C₂, C₄.₅₁(C₂².D₄), (C₂²×D₄).₁₄₉C₂², C₂.₁₇(C₂².₂₉C₂⁴), C₂.C₄².₁₅₂C₂², C₂.₂₅(C₂².₂₆C₂⁴), C₂.₃₆(C₂².₄₇C₂⁴), C₂.₁₆(C₂².₄₉C₂⁴), (C₄×C₂²⋊C₄)⋊₇₅C₂, (C₂×C₄).₈₃₁(C₂×D₄), (C₂×C₄⋊D₄).₃₃C₂, (C₂×C₄²⋊C₂)⋊₃₁C₂, (C₂×C₄).₁₂₇(C₄○D₄), (C₂×C₄⋊C₄).₂₆₉C₂², C₂².₂₇₇(C₂×C₄○D₄), C₂.₃₅(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₁₆₀C₂², SmallGroup(128,1232)

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

Derived series

C₁ — C₂³ — C₂³.₄₀₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₀₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₀₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₀₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₀₀C₂⁴

Generators and relations for C₂³.₄₀₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=e²=1, d²=ca=ac, f²=b, g²=a, ab=ba, ede=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 580 in 284 conjugacy classes, 104 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, 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×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₄×C₂²⋊C₄, C₄²⋊₈C₄, C₄²⋊₉C₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂×C₄²⋊C₂, C₂×C₄⋊D₄, C₂³.₄₀₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂².D₄, C₂².₂₆C₂⁴, C₂².₂₉C₂⁴, C₂².₄₇C₂⁴, C₂².₄₉C₂⁴, C₂³.₄₀₀C₂⁴

Smallest permutation representation of C₂³.₄₀₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	4
size	1	1	···	1	4	4	8	8	2	···	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₀₀C₂⁴

C₄×C₂²⋊C₄

C₄²⋊₈C₄

C₄²⋊₉C₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂×C₄²⋊C₂

C₂×C₄⋊D₄

C₂²×C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₀₀C₂⁴ ►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	4	0
0	0	0	0	0	4

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

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

2	4	0	0	0	0
3	3	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

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

4	3	0	0	0	0
1	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

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

C₂³.₄₀₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{400}C_2^4

% in TeX

G:=Group("C2^3.400C2^4");

// GroupNames label

G:=SmallGroup(128,1232);

// by ID

G=gap.SmallGroup(128,1232);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,184,675,80]);

// Polycyclic

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

// generators/relations

G = C23.400C24 order 128 = 27

117th central stem extension by C23 of C24

G = C₂³.₄₀₀C₂⁴ order 128 = 2⁷

117^th central stem extension by C₂³ of C₂⁴