C2^3.606C2^4 - GroupNames

G = C₂³.₆₀₆C₂⁴ order 128 = 2⁷

323^rd central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₆₀₆C₂⁴, C₂⁴.₄₀₉C₂³, C₂².₂₈₃2_-¹⁺⁴, C₂².₃₈₀2₊¹⁺⁴, C₂²⋊C₄⋊₁₇D₄, C₄²⋊₈C₄⋊₅₆C₂, C₂³.₂₁₈(C₂×D₄), C₂.₆₄(D₄⋊₆D₄), C₂³⋊₂D₄.₂₄C₂, C₂.₁₁₁(D₄⋊₅D₄), C₂³.₇Q₈⋊₉₄C₂, C₂³.₄Q₈⋊₄₆C₂, C₂³.Q₈⋊₆₆C₂, C₂³.₁₇₇(C₄○D₄), C₂³.₁₁D₄⋊₉₀C₂, C₂³.₁₀D₄⋊₉₀C₂, C₂³.₂₃D₄⋊₉₄C₂, (C₂²×C₄).₈₈₁C₂³, (C₂³×C₄).₁₅₂C₂², (C₂×C₄²).₆₅₈C₂², C₂³.₈Q₈⋊₁₀₉C₂, C₂².₄₁₅(C₂²×D₄), C₂⁴.₃C₂²⋊₈₅C₂, (C₂²×D₄).₂₄₁C₂², C₂⁴.C₂²⋊₁₃₆C₂, C₂³.₈₁C₂³⋊₉₁C₂, C₂.₆₀(C₂².₂₉C₂⁴), C₂.₈₁(C₂².₄₅C₂⁴), C₂.C₄².₃₁₂C₂², C₂.₁₉(C₂².₅₆C₂⁴), C₂.₆₈(C₂².₃₃C₂⁴), C₂.₄₄(C₂².₃₄C₂⁴), (C₂×C₄).₁₀₈(C₂×D₄), (C₂×C₄).₄₃₁(C₄○D₄), (C₂×C₄⋊C₄).₄₁₉C₂², C₂².₄₆₈(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄₀C₂, (C₂×C₂²⋊C₄).₂₇₂C₂², SmallGroup(128,1438)

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

Derived series

C₁ — C₂³ — C₂³.₆₀₆C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂³.₆₀₆C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₀₆C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₀₆C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₀₆C₂⁴

Generators and relations for C₂³.₆₀₆C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=b, e²=ba=ab, 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: 596 in 277 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₂².D₄, C₂³.₆₀₆C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₂₉C₂⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₅₆C₂⁴, C₂³.₆₀₆C₂⁴

Smallest permutation representation of C₂³.₆₀₆C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	4A	···	4N	4O	···	4S
order	1	2	···	2	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	8	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₆₀₆C₂⁴

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×C₂².D₄

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₆₀₆C₂⁴ ►in 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

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

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

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

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

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

C₂³.₆₀₆C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{606}C_2^4

% in TeX

G:=Group("C2^3.606C2^4");

// GroupNames label

G:=SmallGroup(128,1438);

// by ID

G=gap.SmallGroup(128,1438);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,344,758,723,100,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=b,e^2=b*a=a*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 = C23.606C24 order 128 = 27

323rd central stem extension by C23 of C24

G = C₂³.₆₀₆C₂⁴ order 128 = 2⁷

323^rd central stem extension by C₂³ of C₂⁴