C2^3.589C2^4 - GroupNames

G = C₂³.₅₈₉C₂⁴ order 128 = 2⁷

306^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₅₈₉C₂⁴, C₂⁴.₃₉₆C₂³, C₂².₃₆₃2₊¹⁺⁴, C₂².₂₆₉2_-¹⁺⁴, C₂²⋊C₄.₁₃D₄, C₂³.₂₁₁(C₂×D₄), C₂.₆₀(D₄⋊₆D₄), C₂.₉₄(D₄⋊₅D₄), (C₂³×C₄).₄₅₄C₂², (C₂²×C₄).₈₇₁C₂³, (C₂×C₄²).₆₄₃C₂², C₂².₃₉₈(C₂²×D₄), C₂³.₈Q₈.₄₅C₂, C₂³.₁₁D₄.₃₃C₂, C₂³.₃₄D₄.₂₆C₂, (C₂²×Q₈).₁₈₁C₂², C₂³.₈₃C₂³⋊₇₆C₂, C₂³.₆₇C₂³⋊₇₈C₂, C₂³.₈₁C₂³⋊₈₀C₂, C₂³.₇₈C₂³⋊₄₂C₂, C₂⁴.C₂².₄₈C₂, C₂³.₆₅C₂³⋊₁₁₈C₂, C₂³.₆₃C₂³⋊₁₂₉C₂, C₂.C₄².₂₉₆C₂², C₂.₅₇(C₂³.₃₈C₂³), C₂.₁₁(C₂².₅₇C₂⁴), C₂.₆₀(C₂².₃₃C₂⁴), C₂.₅₄(C₂².₅₀C₂⁴), C₂.₃₈(C₂².₃₅C₂⁴), (C₂×C₄).₉₄(C₂×D₄), (C₂×C₂²⋊Q₈).₄₂C₂, (C₂×C₄).₁₉₁(C₄○D₄), (C₂×C₄⋊C₄).₄₀₃C₂², C₂².₄₅₁(C₂×C₄○D₄), (C₂×C₄²⋊₂C₂).₁₂C₂, (C₂×C₂²⋊C₄).₂₅₆C₂², SmallGroup(128,1421)

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

Derived series

C₁ — C₂³ — C₂³.₅₈₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂³.₅₈₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₈₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₈₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₈₉C₂⁴

Generators and relations for C₂³.₅₈₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=f²=a, e²=ba=ab, ac=ca, ede^-1=ad=da, geg=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=abd, fg=gf >

Subgroups: 420 in 228 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄²⋊₂C₂, C₂³×C₄, C₂²×Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂×C₄²⋊₂C₂, C₂³.₅₈₉C₂⁴
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₈C₂³, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₅₈₉C₂⁴

Smallest permutation representation of C₂³.₅₈₉C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

(2 44)(4 42)(5 20)(6 8)(7 18)(10 60)(12 58)(14 28)(16 26)(17 19)(21 23)(22 35)(24 33)(30 48)(32 46)(34 36)(37 56)(38 40)(39 54)(49 51)(50 62)(52 64)(53 55)(61 63)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₅₈₉C₂⁴

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₆₇C₂³

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₂²⋊Q₈

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₅₈₉C₂⁴ ►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

4	2	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	1
0	0	0	0	3	4

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

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

1	0	0	0	0	0
1	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	3	4

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

C₂³.₅₈₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{589}C_2^4

% in TeX

G:=Group("C2^3.589C2^4");

// GroupNames label

G:=SmallGroup(128,1421);

// by ID

G=gap.SmallGroup(128,1421);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C23.589C24 order 128 = 27

306th central stem extension by C23 of C24

G = C₂³.₅₈₉C₂⁴ order 128 = 2⁷

306^th central stem extension by C₂³ of C₂⁴