C2^4.259C2^3 - GroupNames

G = C₂⁴.₂₅₉C₂³ order 128 = 2⁷

99^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₂₅₉C₂³, C₂³.₃₂₆C₂⁴, C₂².₁₃₉2₊¹⁺⁴, C₂².₁₀₀2_-¹⁺⁴, (C₂×Q₈)⋊₂₃D₄, C₄⋊C₄.₃₁₉D₄, C₄.₃₀(C₄⋊D₄), C₂.₉(Q₈⋊₆D₄), C₂.₂₆(D₄⋊₅D₄), C₂³.Q₈⋊₇C₂, C₂.₁₃(Q₈⋊₅D₄), C₂³.₂₄(C₄○D₄), C₂³.₁₀D₄⋊₁₇C₂, (C₂²×C₄).₅₀₇C₂³, (C₂³×C₄).₃₃₉C₂², (C₂×C₄²).₄₇₃C₂², C₂².₂₀₆(C₂²×D₄), C₂⁴.C₂²⋊₃₆C₂, C₂⁴.₃C₂²⋊₃₃C₂, (C₂²×D₄).₁₂₄C₂², (C₂²×Q₈).₄₁₈C₂², C₂³.₆₅C₂³⋊₄₃C₂, C₂.C₄².₈₇C₂², C₂.₁₄(C₂².₃₆C₂⁴), C₂.₂₂(C₂³.₃₆C₂³), (C₂×C₄×Q₈)⋊₁₅C₂, (C₂×C₄).₄₉(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₆C₂, (C₄×C₂²⋊C₄)⋊₅₂C₂, (C₂×C₄.₄D₄)⋊₄C₂, C₂.₂₄(C₂×C₄⋊D₄), (C₂×C₄⋊D₄).₂₅C₂, (C₂×C₄).₆₅₃(C₄○D₄), (C₂×C₄⋊C₄).₂₁₃C₂², C₂².₂₀₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₁₆C₂², SmallGroup(128,1158)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₅₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₂₅₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₅₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₅₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₅₉C₂³

Generators and relations for C₂⁴.₂₅₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=e²=b, g²=a, ab=ba, ac=ca, ede^-1=gdg^-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 628 in 316 conjugacy classes, 112 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×Q₈, C₄⋊D₄, C₂²⋊Q₈, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂×C₄×Q₈, C₂×C₄⋊D₄, C₂×C₂²⋊Q₈, C₂×C₄.₄D₄, C₂⁴.₂₅₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₄⋊D₄, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, Q₈⋊₅D₄, Q₈⋊₆D₄, C₂⁴.₂₅₉C₂³

Smallest permutation representation of C₂⁴.₂₅₉C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	4
size	1	1	···	1	4	4	8	8	2	···	2	4	···	4	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₂₅₉C₂³

C₄×C₂²⋊C₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂×C₄×Q₈

C₂×C₄⋊D₄

C₂×C₂²⋊Q₈

C₂×C₄.₄D₄

C₄⋊C₄

C₂×Q₈

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₅₉C₂³ ►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

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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

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

C₂⁴.₂₅₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{259}C_2^3

% in TeX

G:=Group("C2^4.259C2^3");

// GroupNames label

G:=SmallGroup(128,1158);

// by ID

G=gap.SmallGroup(128,1158);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,675,80]);

// Polycyclic

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

// generators/relations

G = C24.259C23 order 128 = 27

99th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₅₉C₂³ order 128 = 2⁷

99^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²