C2^3.630C2^4 - GroupNames

G = C₂³.₆₃₀C₂⁴ order 128 = 2⁷

347^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₃₀C₂⁴, C₂⁴.₄₂₂C₂³, C₂².₄₀₃2₊¹⁺⁴, C₂².₃₀₅2_-¹⁺⁴, (C₂×Q₈)⋊₁₃D₄, C₂.₅₆(Q₈⋊₅D₄), C₂.₃₄(Q₈⋊₆D₄), C₂³.₈₆(C₄○D₄), C₂³⋊₂D₄.₂₇C₂, C₂³.Q₈⋊₇₂C₂, C₂³.₄Q₈⋊₅₂C₂, C₂³.₁₀D₄⋊₉₇C₂, C₂.₅₄(C₂³⋊₃D₄), (C₂²×C₄).₁₉₉C₂³, (C₂³×C₄).₄₇₇C₂², (C₂×C₄²).₆₈₁C₂², C₂².₄₃₉(C₂²×D₄), C₂³.₁₁D₄⋊₁₀₁C₂, C₂³.₂₃D₄⋊₁₀₀C₂, C₂⁴.₃C₂²⋊₈₉C₂, (C₂²×D₄).₂₅₆C₂², (C₂²×Q₈).₁₉₉C₂², C₂³.₆₇C₂³⋊₉₁C₂, C₂⁴.C₂²⋊₁₄₇C₂, C₂.₇₅(C₂².₃₂C₂⁴), C₂³.₆₃C₂³⋊₁₅₁C₂, C₂.₈₆(C₂².₄₅C₂⁴), C₂.C₄².₃₃₆C₂², C₂.₂₃(C₂².₅₆C₂⁴), C₂.₈₆(C₂².₃₆C₂⁴), (C₂×C₄).₁₂₄(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₄₅C₂, (C₂×C₄.₄D₄)⋊₃₁C₂, (C₂×C₄).₂₁₀(C₄○D₄), (C₂×C₄⋊C₄).₄₄₃C₂², C₂².₄₉₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₉₃C₂², SmallGroup(128,1462)

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

Derived series

C₁ — C₂³ — C₂³.₆₃₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.C₂² — C₂³.₆₃₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₃₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₃₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₃₀C₂⁴

Generators and relations for C₂³.₆₃₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=g²=cb=bc, f²=b, eae^-1=gag^-1=ab=ba, ac=ca, faf^-1=ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=bce, fg=gf >

Subgroups: 596 in 274 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, 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, C₂²⋊Q₈, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₄Q₈, C₂×C₂²⋊Q₈, C₂×C₄.₄D₄, C₂³.₆₃₀C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³⋊₃D₄, C₂².₃₂C₂⁴, C₂².₃₆C₂⁴, Q₈⋊₅D₄, Q₈⋊₆D₄, C₂².₄₅C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₃₀C₂⁴

Smallest permutation representation of C₂³.₆₃₀C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	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

C₂³.₆₃₀C₂⁴

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₆₇C₂³

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₄Q₈

C₂×C₂²⋊Q₈

C₂×C₄.₄D₄

C₂×Q₈

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₃₀C₂⁴ ►in GL₆(𝔽₅)

2	2	0	0	0	0
1	3	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	1	0

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

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	0	2	0	0
0	0	2	0	0	0
0	0	0	0	3	0
0	0	0	0	0	2

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

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	0	1
0	0	0	0	4	0

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

C₂³.₆₃₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{630}C_2^4

% in TeX

G:=Group("C2^3.630C2^4");

// GroupNames label

G:=SmallGroup(128,1462);

// by ID

G=gap.SmallGroup(128,1462);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23.630C24 order 128 = 27

347th central stem extension by C23 of C24

G = C₂³.₆₃₀C₂⁴ order 128 = 2⁷

347^th central stem extension by C₂³ of C₂⁴