C2^3.404C2^4 - GroupNames

G = C₂³.₄₀₄C₂⁴ order 128 = 2⁷

121^st central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₁₅C₂³, C₂³.₄₀₄C₂⁴, C₂².₂₀₀2₊¹⁺⁴, C₄²⋊₈C₄⋊₃₂C₂, (C₂²×C₄).₃₉₁D₄, C₂³.₆₁₈(C₂×D₄), C₂³.₃₃₀(C₄○D₄), C₂³.₁₁D₄⋊₃₁C₂, C₂³.₂₃D₄⋊₅₁C₂, (C₂×C₄²).₅₂₄C₂², (C₂²×C₄).₅₂₅C₂³, (C₂³×C₄).₁₀₁C₂², C₂².₂₈₀(C₂²×D₄), C₂⁴.₃C₂²⋊₅₁C₂, C₂².₃₉(C₄.₄D₄), C₄.₅₄(C₂².D₄), (C₂²×D₄).₁₅₁C₂², C₂.₁₉(C₂².₂₉C₂⁴), C₂.C₄².₁₅₅C₂², C₂.₃₇(C₂².₄₇C₂⁴), (C₄×C₂²⋊C₄)⋊₇₇C₂, (C₂²×C₄⋊C₄)⋊₂₆C₂, (C₂×C₄).₈₃₃(C₂×D₄), (C₂×C₄⋊D₄).₃₅C₂, C₂.₂₀(C₂×C₄.₄D₄), (C₂×C₄).₈₁₆(C₄○D₄), (C₂×C₄⋊C₄).₈₆₀C₂², C₂².₂₈₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₇C₂², C₂.₃₉(C₂×C₂².D₄), SmallGroup(128,1236)

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

Derived series

C₁ — C₂³ — C₂³.₄₀₄C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₀₄C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₀₄C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₀₄C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₀₄C₂⁴

Generators and relations for C₂³.₄₀₄C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ca=ac, e²=g²=a, f²=ba=ab, ede^-1=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: 612 in 298 conjugacy classes, 108 normal (16 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×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₄×C₂²⋊C₄, C₄²⋊₈C₄, C₂³.₂₃D₄, C₂⁴.₃C₂², C₂³.₁₁D₄, C₂²×C₄⋊C₄, C₂×C₄⋊D₄, C₂³.₄₀₄C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×C₂².D₄, C₂×C₄.₄D₄, C₂².₂₉C₂⁴, C₂².₄₇C₂⁴, C₂³.₄₀₄C₂⁴

Smallest permutation representation of C₂³.₄₀₄C₂⁴
►On 64 points

Generators in S₆₄

(1 36)(2 33)(3 34)(4 35)(5 63)(6 64)(7 61)(8 62)(9 16)(10 13)(11 14)(12 15)(17 22)(18 23)(19 24)(20 21)(25 32)(26 29)(27 30)(28 31)(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 56)(18 53)(19 54)(20 55)(21 50)(22 51)(23 52)(24 49)(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 14)(10 15)(11 16)(12 13)(17 24)(18 21)(19 22)(20 23)(25 30)(26 31)(27 32)(28 29)(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 54 16 49)(10 50 13 55)(11 56 14 51)(12 52 15 53)(17 28 22 31)(18 32 23 25)(19 26 24 29)(20 30 21 27)

(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 53 26 23)(10 19 27 49)(11 55 28 21)(12 17 25 51)(13 24 30 54)(14 50 31 20)(15 22 32 56)(16 52 29 18)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4V	4W	4X
order	1	2	···	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4
size	1	1	···	1	2	2	2	2	8	8	2	2	2	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂³.₄₀₄C₂⁴

C₄×C₂²⋊C₄

C₄²⋊₈C₄

C₂³.₂₃D₄

C₂⁴.₃C₂²

C₂³.₁₁D₄

C₂²×C₄⋊C₄

C₂×C₄⋊D₄

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₄₀₄C₂⁴ ►in GL₆(𝔽₅)

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	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

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

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

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

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	0	1
0	0	0	0	4	0

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

C₂³.₄₀₄C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{404}C_2^4

% in TeX

G:=Group("C2^3.404C2^4");

// GroupNames label

G:=SmallGroup(128,1236);

// by ID

G=gap.SmallGroup(128,1236);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,268,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*a=a*c,e^2=g^2=a,f^2=b*a=a*b,e*d*e^-1=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 = C23.404C24 order 128 = 27

121st central stem extension by C23 of C24

G = C₂³.₄₀₄C₂⁴ order 128 = 2⁷

121^st central stem extension by C₂³ of C₂⁴