C2^3.677C2^4

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

Aliases: C₂⁴.₈₇C₂³, C₂³.₆₇₇C₂⁴, C₂².₄₅₀2₊⁽¹⁺⁴⁾, C₂².₃₄₃2_-⁽¹⁺⁴⁾, C₄²⋊₅C₄⋊₃₆C₂, (C₂²×C₄).₅₉₂C₂³, (C₂×C₄²).₇₀₈C₂², C₂³.₄Q₈.₂₈C₂, C₂³.Q₈.₃₉C₂, C₂³.₁₁D₄.₅₄C₂, C₂⁴.C₂².₇₄C₂, C₂³.₈₃C₂³⋊₁₁₃C₂, C₂³.₈₁C₂³⋊₁₂₂C₂, C₂³.₆₅C₂³⋊₁₅₀C₂, C₂³.₆₃C₂³⋊₁₇₉C₂, C₂.C₄².₃₈₁C₂², C₂.₆₇(C₂².₅₀C₂⁴), C₂.₄₈(C₂².₅₇C₂⁴), C₂.₆₀(C₂².₃₄C₂⁴), C₂.₁₁₅(C₂².₄₆C₂⁴), C₂.₁₀₀(C₂².₃₃C₂⁴), C₂.₁₀₆(C₂².₄₇C₂⁴), C₂.₁₁₄(C₂².₃₆C₂⁴), (C₂×C₄).₂₂₆(C₄○D₄), (C₂×C₄⋊C₄).₄₈₇C₂², C₂².₅₃₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₁₅C₂², SmallGroup(128,1509)

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

Derived series

C₁ — C₂³ — C₂³.₆₇₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₅C₂³ — C₂³.₆₇₇C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₇₇C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₇₇C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₇₇C₂⁴

Subgroups: 356 in 189 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂, C₄^[×17], C₂²^[×7], C₂²^[×7], C₂×C₄^[×6], C₂×C₄^[×39], C₂³, C₂³^[×7], C₄²^[×4], C₂²⋊C₄^[×10], C₄⋊C₄^[×16], C₂²×C₄^[×14], C₂⁴, C₂.C₄²^[×14], C₂×C₄²^[×3], C₂×C₂²⋊C₄^[×7], C₂×C₄⋊C₄^[×11], C₄²⋊₅C₄, C₂³.₆₃C₂³^[×2], C₂⁴.C₂²^[×3], C₂³.₆₅C₂³^[×3], C₂³.Q₈, C₂³.₁₁D₄^[×2], C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₆₇₇C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×2], 2_-⁽¹⁺⁴⁾^[×2], C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₇₇C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=abc, e²=f²=ba=ab, g²=a, ac=ca, ede^-1=ad=da, geg^-1=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, gdg^-1=abd, fg=gf >

Smallest permutation representation
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ 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	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	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	3	0	0
0	0	1	1	0	0
0	0	0	0	0	1
0	0	0	0	4	0

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

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

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

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

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

C₂³.₆₇₇C₂⁴

C₄²⋊₅C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^3._{677}C_2^4

% in TeX

G:=Group("C2^3.677C2^4");

// GroupNames label

G:=SmallGroup(128,1509);

// by ID

G=gap.SmallGroup(128,1509);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,268,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C₂³.₆₇₇C₂⁴ order 128 = 2⁷

394^th central stem extension by C₂³ of C₂⁴

G = C23.677C24 order 128 = 27

394th central stem extension by C23 of C24

G = C₂³.₆₇₇C₂⁴ order 128 = 2⁷

394^th central stem extension by C₂³ of C₂⁴