C2^3.673C2^4 - GroupNames

G = C₂³.₆₇₃C₂⁴ order 128 = 2⁷

390^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₈₅C₂³, C₂³.₆₇₃C₂⁴, C₂².₃₃₉2_-¹⁺⁴, C₂².₄₄₆2₊¹⁺⁴, C₄²⋊₈C₄⋊₆₆C₂, (C₂²×C₄).₅₉₀C₂³, (C₂×C₄²).₇₀₄C₂², C₂³.₄Q₈.₂₇C₂, C₂³.₁₁D₄.₅₂C₂, C₂⁴.C₂².₇₂C₂, C₂³.₈₁C₂³⋊₁₂₁C₂, C₂³.₆₃C₂³⋊₁₇₇C₂, C₂³.₆₅C₂³⋊₁₄₈C₂, C₂³.₈₃C₂³⋊₁₁₁C₂, C₂.C₄².₃₇₇C₂², C₂.₉₇(C₂².₃₃C₂⁴), C₂.₄₁(C₂².₅₃C₂⁴), C₂.₅₉(C₂².₃₄C₂⁴), C₂.₄₄(C₂².₅₇C₂⁴), C₂.₁₁₁(C₂².₄₆C₂⁴), C₂.₁₁₁(C₂².₃₆C₂⁴), (C₂×C₄).₂₂₃(C₄○D₄), (C₂×C₄⋊C₄).₄₈₃C₂², C₂².₅₃₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₄C₂², SmallGroup(128,1505)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₃C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₇₃C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₃C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₃C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₃C₂⁴

Generators and relations for C₂³.₆₇₃C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=cb=bc, e²=ba=ab, f²=b, g²=a, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 356 in 189 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₈C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₇₃C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₅₃C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₇₃C₂⁴

Smallest permutation representation of C₂³.₆₇₃C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	4A	···	4R	4S	···	4W
order	1	2	···	2	2	4	···	4	4	···	4
size	1	1	···	1	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₇₃C₂⁴

C₄²⋊₈C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₇₃C₂⁴ ►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	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	1	0
0	0	0	0	0	1

0	1	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	0	4
0	0	0	0	1	0

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

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

2	0	0	0	0	0
0	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

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

C₂³.₆₇₃C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{673}C_2^4

% in TeX

G:=Group("C2^3.673C2^4");

// GroupNames label

G:=SmallGroup(128,1505);

// by ID

G=gap.SmallGroup(128,1505);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,344,758,723,184,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C23.673C24 order 128 = 27

390th central stem extension by C23 of C24

G = C₂³.₆₇₃C₂⁴ order 128 = 2⁷

390^th central stem extension by C₂³ of C₂⁴