C2^3.605C2^4 - GroupNames

G = C₂³.₆₀₅C₂⁴ order 128 = 2⁷

322^nd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₆₅C₂³, C₂³.₆₀₅C₂⁴, C₂².₂₈₂2_-¹⁺⁴, C₂².₃₇₉2₊¹⁺⁴, C₄⋊C₄⋊₁₆D₄, C₂.₃₀(Q₈⋊₆D₄), C₂³⋊₂D₄.₂₃C₂, C₂.₁₁₀(D₄⋊₅D₄), C₂³.₄Q₈⋊₄₅C₂, C₂³.₇Q₈⋊₉₃C₂, C₂³.Q₈⋊₆₅C₂, C₂³.₁₇₆(C₄○D₄), C₂³.₂₃D₄⋊₉₃C₂, C₂³.₁₀D₄⋊₈₉C₂, (C₂×C₄²).₆₅₇C₂², (C₂³×C₄).₄₆₅C₂², (C₂²×C₄).₈₈₀C₂³, C₂².₄₁₄(C₂²×D₄), C₂⁴.₃C₂²⋊₈₄C₂, (C₂²×D₄).₂₄₀C₂², C₂⁴.C₂²⋊₁₃₅C₂, C₂³.₆₃C₂³⋊₁₃₇C₂, C₂³.₆₅C₂³⋊₁₂₅C₂, C₂.₁₈(C₂².₅₄C₂⁴), C₂.₈₀(C₂².₄₅C₂⁴), C₂.C₄².₃₁₁C₂², C₂.₄₈(C₂².₃₁C₂⁴), C₂.₇₇(C₂².₃₆C₂⁴), C₂.₆₇(C₂².₃₃C₂⁴), (C₂×C₄).₁₀₇(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₂₁C₂, (C₂×C₄).₁₉₄(C₄○D₄), (C₂×C₄⋊C₄).₄₁₈C₂², C₂².₄₆₇(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₃₉C₂, (C₂×C₂²⋊C₄).₂₇₁C₂², SmallGroup(128,1437)

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

Derived series

C₁ — C₂³ — C₂³.₆₀₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂³.₆₀₅C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₀₅C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₀₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₀₅C₂⁴

Generators and relations for C₂³.₆₀₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=ba=ab, e²=b, f²=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 >

Subgroups: 580 in 270 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₄Q₈, C₂×C₂².D₄, C₂×C₄²⋊₂C₂, C₂³.₆₀₅C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₁C₂⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, D₄⋊₅D₄, Q₈⋊₆D₄, C₂².₄₅C₂⁴, C₂².₅₄C₂⁴, C₂³.₆₀₅C₂⁴

Smallest permutation representation of C₂³.₆₀₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

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

C₂³.₆₀₅C₂⁴

C₂³.₇Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₄Q₈

C₂×C₂².D₄

C₂×C₄²⋊₂C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₀₅C₂⁴ ►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

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

4	1	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	2	0
0	0	0	0	0	2

2	0	0	0	0	0
0	2	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	0	0	0	4
0	0	0	0	4	0

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

C₂³.₆₀₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{605}C_2^4

% in TeX

G:=Group("C2^3.605C2^4");

// GroupNames label

G:=SmallGroup(128,1437);

// by ID

G=gap.SmallGroup(128,1437);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,100,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=b*a=a*b,e^2=b,f^2=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 = C23.605C24 order 128 = 27

322nd central stem extension by C23 of C24

G = C₂³.₆₀₅C₂⁴ order 128 = 2⁷

322^nd central stem extension by C₂³ of C₂⁴