C2^4.331C2^3 - GroupNames

G = C₂⁴.₃₃₁C₂³ order 128 = 2⁷

171^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₃₁C₂³, C₂³.₄₅₉C₂⁴, C₂².₂₄₄2₊¹⁺⁴, C₂²⋊C₄.₇₇D₄, C₂³.₅₅(C₂×D₄), C₂.₇₉(D₄⋊₅D₄), C₂³.₅₂(C₄○D₄), (C₂×C₄²).₆₄C₂², C₂³.₈Q₈⋊₆₇C₂, C₂³.₁₀D₄⋊₄₅C₂, C₂³.₂₃D₄⋊₅₉C₂, (C₂²×C₄).₅₄₁C₂³, (C₂³×C₄).₁₁₇C₂², C₂².₃₁₀(C₂²×D₄), C₂⁴.C₂²⋊₈₅C₂, C₂⁴.₃C₂²⋊₆₀C₂, (C₂²×D₄).₁₇₃C₂², C₂³.₆₅C₂³⋊₈₉C₂, C₂³.₈₃C₂³⋊₃₈C₂, C₂.₂₅(C₂².₃₂C₂⁴), C₂.C₄².₁₉₆C₂², C₂.₃₉(C₂².₂₆C₂⁴), C₂.₈₁(C₂³.₃₆C₂³), C₂.₅₆(C₂².₄₇C₂⁴), (C₄×C₂²⋊C₄)⋊₈₉C₂, (C₂×C₄).₉₀₉(C₂×D₄), (C₂×C₄⋊D₄).₃₆C₂, (C₂×C₄²⋊₂C₂)⋊₁₁C₂, (C₂×C₄).₃₉₀(C₄○D₄), (C₂×C₄⋊C₄).₃₁₀C₂², C₂².₃₃₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₀₈C₂², SmallGroup(128,1291)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₃₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₃₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₃₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₃₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₃₁C₂³

Generators and relations for C₂⁴.₃₃₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=cb=bc, g²=ba=ab, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 596 in 287 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.₈₃C₂³, C₂×C₄⋊D₄, C₂×C₄²⋊₂C₂, C₂⁴.₃₃₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₂C₂⁴, D₄⋊₅D₄, C₂².₄₇C₂⁴, C₂⁴.₃₃₁C₂³

Smallest permutation representation of C₂⁴.₃₃₁C₂³
►On 64 points

Generators in S₆₄

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

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

(1 7)(2 8)(3 5)(4 6)(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)(33 49)(34 50)(35 51)(36 52)

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₃₃₁C₂³

C₄×C₂²⋊C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.₈₃C₂³

C₂×C₄⋊D₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₃₁C₂³ ►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	4	0
0	0	0	0	0	4

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

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

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

0	1	0	0	0	0
1	0	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	4	0

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

C₂⁴.₃₃₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{331}C_2^3

% in TeX

G:=Group("C2^4.331C2^3");

// GroupNames label

G:=SmallGroup(128,1291);

// by ID

G=gap.SmallGroup(128,1291);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,680,758,723,675,192]);

// Polycyclic

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

// generators/relations

G = C24.331C23 order 128 = 27

171st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₃₁C₂³ order 128 = 2⁷

171^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²