C2^3.580C2^4 - GroupNames

G = C₂³.₅₈₀C₂⁴ order 128 = 2⁷

297^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₅₈₀C₂⁴, C₂⁴.₃₈₈C₂³, C₂².₂₆₃2_-¹⁺⁴, C₂².₃₅₄2₊¹⁺⁴, C₂.₄₄D₄², C₂²⋊C₄.₉D₄, C₄²⋊₉C₄⋊₃₂C₂, C₂³.₂₀₆(C₂×D₄), C₂.₅₇(D₄⋊₆D₄), C₂³.₈Q₈⋊₉₇C₂, C₂³.₁₁D₄⋊₇₆C₂, (C₂²×C₄).₈₆₇C₂³, (C₂×C₄²).₆₃₇C₂², (C₂³×C₄).₄₄₈C₂², C₂².₃₈₉(C₂²×D₄), C₂³.₁₀D₄.₃₉C₂, (C₂²×D₄).₂₁₉C₂², (C₂²×Q₈).₁₇₆C₂², C₂⁴.C₂²⋊₁₂₀C₂, C₂³.₈₁C₂³⋊₇₇C₂, C₂³.₇₈C₂³⋊₃₉C₂, C₂.C₄².₂₈₉C₂², C₂.₉(C₂².₅₇C₂⁴), C₂.₅₇(C₂².₃₃C₂⁴), C₂.₅₄(C₂³.₃₈C₂³), (C₂×C₄).₈₈(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₃₅C₂, (C₂×C₄).₄₁₇(C₄○D₄), (C₂×C₄⋊C₄).₃₉₆C₂², C₂².₄₄₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₄₉C₂², (C₂×C₂².D₄).₂₃C₂, SmallGroup(128,1412)

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

Derived series

C₁ — C₂³ — C₂³.₅₈₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂×C₂².D₄ — C₂³.₅₈₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₈₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₈₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₈₀C₂⁴

Generators and relations for C₂³.₅₈₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=c, g²=b, faf=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bc=cb, bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 548 in 284 conjugacy classes, 104 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₄²⋊₉C₄, C₂³.₈Q₈, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂×C₂²⋊Q₈, C₂×C₂².D₄, C₂³.₅₈₀C₂⁴
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, D₄², D₄⋊₆D₄, C₂².₅₇C₂⁴, C₂³.₅₈₀C₂⁴

Smallest permutation representation of C₂³.₅₈₀C₂⁴
►On 64 points

Generators in S₆₄

(2 38)(4 40)(5 17)(6 8)(7 19)(10 44)(12 42)(13 45)(14 16)(15 47)(18 20)(22 52)(24 50)(26 56)(28 54)(29 57)(30 32)(31 59)(33 35)(34 63)(36 61)(46 48)(58 60)(62 64)

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₅₈₀C₂⁴

C₄²⋊₉C₄

C₂³.₈Q₈

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₂²⋊Q₈

C₂×C₂².D₄

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₅₈₀C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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

C₂³.₅₈₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{580}C_2^4

% in TeX

G:=Group("C2^3.580C2^4");

// GroupNames label

G:=SmallGroup(128,1412);

// by ID

G=gap.SmallGroup(128,1412);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23.580C24 order 128 = 27

297th central stem extension by C23 of C24

G = C₂³.₅₈₀C₂⁴ order 128 = 2⁷

297^th central stem extension by C₂³ of C₂⁴