C2^3.590C2^4 - GroupNames

G = C₂³.₅₉₀C₂⁴ order 128 = 2⁷

307^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂³.₅₉₀C₂⁴, C₂⁴.₃₉₇C₂³, C₂².₃₆₄2₊¹⁺⁴, C₂².₂₇₀2_-¹⁺⁴, C₄⋊C₄.₁₁₇D₄, C₂.₉₅(D₄⋊₅D₄), C₂.₄₅(Q₈⋊₅D₄), C₂³.₇₇(C₄○D₄), (C₂²×C₄).₁₈₁C₂³, (C₂³×C₄).₄₅₅C₂², (C₂×C₄²).₆₄₄C₂², C₂².₃₉₉(C₂²×D₄), C₂³.₈Q₈.₄₆C₂, C₂³.₇Q₈.₆₄C₂, C₂³.₁₁D₄.₃₄C₂, (C₂²×Q₈).₁₈₂C₂², C₂³.₈₃C₂³⋊₇₇C₂, C₂³.₆₇C₂³⋊₇₉C₂, C₂³.₈₁C₂³⋊₈₁C₂, C₂³.₇₈C₂³⋊₄₃C₂, C₂⁴.C₂².₄₉C₂, C₂³.₆₃C₂³⋊₁₃₀C₂, C₂.C₄².₂₉₇C₂², C₂.₆₁(C₂².₃₃C₂⁴), C₂.₈₁(C₂².₄₆C₂⁴), C₂.₁₂(C₂².₅₇C₂⁴), C₂.₃₉(C₂².₃₅C₂⁴), C₂.₅₈(C₂³.₃₈C₂³), (C₂×C₄).₉₅(C₂×D₄), (C₂×C₄².C₂)⋊₂₀C₂, (C₂×C₂²⋊Q₈).₄₃C₂, (C₂×C₄).₄₂₁(C₄○D₄), (C₂×C₄⋊C₄).₄₀₄C₂², C₂².₄₅₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₂₅₇C₂², SmallGroup(128,1422)

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

Derived series

C₁ — C₂³ — C₂³.₅₉₀C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂³.₇Q₈ — C₂³.₅₉₀C₂⁴

Lower central

C₁ — C₂³ — C₂³.₅₉₀C₂⁴

Upper central

C₁ — C₂³ — C₂³.₅₉₀C₂⁴

Jennings

C₁ — C₂³ — C₂³.₅₉₀C₂⁴

Generators and relations for C₂³.₅₉₀C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=g²=a, e²=b, ab=ba, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg^-1=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, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₄².C₂, C₂³×C₄, C₂²×Q₈, C₂³.₇Q₈, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂×C₂²⋊Q₈, C₂×C₄².C₂, C₂³.₅₉₀C₂⁴
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, Q₈⋊₅D₄, C₂².₄₆C₂⁴, C₂².₅₇C₂⁴, C₂³.₅₉₀C₂⁴

Smallest permutation representation of C₂³.₅₉₀C₂⁴
►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 49 41 61)(2 52 42 64)(3 51 43 63)(4 50 44 62)(5 58 20 10)(6 57 17 9)(7 60 18 12)(8 59 19 11)(13 34 25 21)(14 33 26 24)(15 36 27 23)(16 35 28 22)(29 53 45 38)(30 56 46 37)(31 55 47 40)(32 54 48 39)

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

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

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

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

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

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

C₂³.₅₉₀C₂⁴

C₂³.₇Q₈

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₂²⋊Q₈

C₂×C₄².C₂

C₄⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂³.₅₉₀C₂⁴ ►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

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

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

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

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

C₂³.₅₉₀C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{590}C_2^4

% in TeX

G:=Group("C2^3.590C2^4");

// GroupNames label

G:=SmallGroup(128,1422);

// by ID

G=gap.SmallGroup(128,1422);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,1571,346,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=g^2=a,e^2=b,a*b=b*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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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 = C23.590C24 order 128 = 27

307th central stem extension by C23 of C24

G = C₂³.₅₉₀C₂⁴ order 128 = 2⁷

307^th central stem extension by C₂³ of C₂⁴