C2^3.589C2^4

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

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

Subgroups: 420 in 228 conjugacy classes, 96 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×2], C₄^[×18], C₂²^[×7], C₂²^[×10], C₂×C₄^[×8], C₂×C₄^[×42], Q₈^[×4], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×3], C₂²⋊C₄^[×4], C₂²⋊C₄^[×9], C₄⋊C₄^[×20], C₂²×C₄^[×14], C₂²×C₄^[×5], C₂×Q₈^[×5], C₂⁴, C₂.C₄²^[×14], C₂×C₄²^[×2], C₂×C₂²⋊C₄^[×6], C₂×C₄⋊C₄^[×11], C₂²⋊Q₈^[×4], C₄²⋊₂C₂^[×4], C₂³×C₄, C₂²×Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³^[×2], C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₁₁D₄^[×2], C₂³.₈₁C₂³, C₂³.₈₃C₂³^[×2], C₂×C₂²⋊Q₈, C₂×C₄²⋊₂C₂, C₂³.₅₈₉C₂⁴

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×4], C₂⁴, C₂²×D₄, C₂×C₄○D₄^[×2], 2₊⁽¹⁺⁴⁾, 2_-⁽¹⁺⁴⁾^[×3], C₂³.₃₈C₂³, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₅₈₉C₂⁴

Generators and relations
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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

306^th central stem extension by C₂³ of C₂⁴

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