C2^3.631C2^4 - GroupNames

G = C₂³.₆₃₁C₂⁴ order 128 = 2⁷

348^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₇₄C₂³, C₂³.₆₃₁C₂⁴, C₂².₄₀₄2₊¹⁺⁴, C₂².₃₀₆2_-¹⁺⁴, (C₂×Q₈).₁₂₁D₄, C₂³⋊Q₈⋊₅₁C₂, C₂.₅₇(Q₈⋊₅D₄), C₂³.Q₈⋊₇₃C₂, C₂.₅₅(C₂³⋊₃D₄), (C₂²×C₄).₂₀₀C₂³, (C₂×C₄²).₆₈₂C₂², C₂².₄₄₀(C₂²×D₄), C₂³.₁₀D₄.₅₁C₂, (C₂²×D₄).₂₅₇C₂², (C₂²×Q₈).₂₀₀C₂², C₂³.₆₇C₂³⋊₉₂C₂, C₂⁴.C₂²⋊₁₄₈C₂, C₂³.₆₃C₂³⋊₁₅₂C₂, C₂.C₄².₃₃₇C₂², C₂.₃₆(C₂².₄₉C₂⁴), C₂.₈₇(C₂².₃₆C₂⁴), C₂.₃₁(C₂².₅₇C₂⁴), (C₂×C₄⋊Q₈)⋊₂₄C₂, (C₂×C₄).₁₂₅(C₂×D₄), (C₂×C₄).₂₁₁(C₄○D₄), (C₂×C₄⋊C₄).₄₄₄C₂², (C₂×C₄.₄D₄).₃₄C₂, C₂².₄₉₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₄C₂², SmallGroup(128,1463)

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

Derived series

C₁ — C₂³ — C₂³.₆₃₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₃C₂³ — C₂³.₆₃₁C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₃₁C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₃₁C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₃₁C₂⁴

Generators and relations for C₂³.₆₃₁C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=g²=ba=ab, e²=f²=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 >

Subgroups: 500 in 246 conjugacy classes, 96 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄.₄D₄, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₀D₄, C₂³.Q₈, C₂×C₄.₄D₄, C₂×C₄⋊Q₈, C₂³.₆₃₁C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₆C₂⁴, Q₈⋊₅D₄, C₂².₄₉C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₃₁C₂⁴

Smallest permutation representation of C₂³.₆₃₁C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4R	4S	4T	4U	4V
order	1	2	···	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₃₁C₂⁴

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₀D₄

C₂³.Q₈

C₂×C₄.₄D₄

C₂×C₄⋊Q₈

C₂×Q₈

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₆₃₁C₂⁴ ►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

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

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

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	3	0
0	0	0	0	0	2

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

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

C₂³.₆₃₁C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{631}C_2^4

% in TeX

G:=Group("C2^3.631C2^4");

// GroupNames label

G:=SmallGroup(128,1463);

// by ID

G=gap.SmallGroup(128,1463);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,100,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=b*a=a*b,e^2=f^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 = C23.631C24 order 128 = 27

348th central stem extension by C23 of C24

G = C₂³.₆₃₁C₂⁴ order 128 = 2⁷

348^th central stem extension by C₂³ of C₂⁴