C2^3.455C2^4 - GroupNames

G = C₂³.₄₅₅C₂⁴ order 128 = 2⁷

172^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₄₅₅C₂⁴, C₂⁴.₃₂₈C₂³, C₂².₂₄₀2₊¹⁺⁴, C₂.₃₁D₄², C₂²⋊C₄⋊₂₈D₄, C₂³.₅₃(C₂×D₄), C₂³⋊₂D₄⋊₂₁C₂, C₂.₇₅(D₄⋊₅D₄), C₂³.₁₀D₄⋊₄₄C₂, C₂³.₂₃D₄⋊₅₇C₂, (C₂³×C₄).₃₉₉C₂², (C₂×C₄²).₅₆₀C₂², C₂².₃₀₆(C₂²×D₄), C₂⁴.C₂²⋊₈₂C₂, C₂⁴.₃C₂²⋊₅₈C₂, (C₂²×C₄).₁₂₅₉C₂³, (C₂²×D₄).₁₇₀C₂², (C₂²×Q₈).₁₃₅C₂², C₂³.₇₈C₂³⋊₁₆C₂, C₂.₂₃(C₂².₃₂C₂⁴), C₂.C₄².₁₉₂C₂², C₂.₃₅(C₂².₂₆C₂⁴), C₂.₂₅(C₂².₄₉C₂⁴), (C₂×C₄⋊D₄)⋊₂₀C₂, (C₄×C₂²⋊C₄)⋊₈₆C₂, (C₂×C₄).₉₀₇(C₂×D₄), (C₂×C₄.₄D₄)⋊₁₇C₂, (C₂×C₄).₃₈₈(C₄○D₄), (C₂×C₄⋊C₄).₃₀₇C₂², C₂².₃₃₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₀₆C₂², SmallGroup(128,1287)

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

Derived series

C₁ — C₂³ — C₂³.₄₅₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂³.₄₅₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₅₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₅₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₅₅C₂⁴

Generators and relations for C₂³.₄₅₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=ca=ac, f²=a, g²=b, ab=ba, 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: 756 in 348 conjugacy classes, 108 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂×C₄⋊D₄, C₂×C₄.₄D₄, C₂³.₄₅₅C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₂₆C₂⁴, C₂².₃₂C₂⁴, D₄², D₄⋊₅D₄, C₂².₄₉C₂⁴, C₂³.₄₅₅C₂⁴

Smallest permutation representation of C₂³.₄₅₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

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

C₂³.₄₅₅C₂⁴

C₄×C₂²⋊C₄

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂×C₄⋊D₄

C₂×C₄.₄D₄

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₅₅C₂⁴ ►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	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

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	0	3
0	0	0	0	2	0

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

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

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

C₂³.₄₅₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{455}C_2^4

% in TeX

G:=Group("C2^3.455C2^4");

// GroupNames label

G:=SmallGroup(128,1287);

// by ID

G=gap.SmallGroup(128,1287);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,100,675,136]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c*a=a*c,f^2=a,g^2=b,a*b=b*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,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 = C23.455C24 order 128 = 27

172nd central stem extension by C23 of C24

G = C₂³.₄₅₅C₂⁴ order 128 = 2⁷

172^nd central stem extension by C₂³ of C₂⁴