C2^4.598C2^3 - GroupNames

G = C₂⁴.₅₉₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₉₈C₂³, C₂³.₇₅₄C₂⁴, (C₂²×C₄)⋊₄₇D₄, (C₂²×C₄²)⋊₁₆C₂, C₂³.₆₃₃(C₂×D₄), C₄⋊₄(C₂².D₄), C₂³.₂₅₄(C₄○D₄), C₂².₃₁(C₄⋊D₄), (C₂²×C₄).₂₆₂C₂³, (C₂³×C₄).₆₅₄C₂², C₂³.₇Q₈⋊₁₁₇C₂, C₂².₄₆₄(C₂²×D₄), C₂³.₂₃D₄⋊₁₁₄C₂, (C₂×C₄²).₁₀₁₅C₂², (C₂²×D₄).₃₁₂C₂², C₂⁴.C₂²⋊₁₈₅C₂, C₂⁴.₃C₂²⋊₁₀₀C₂, C₂.₉₇(C₂².₁₉C₂⁴), C₂³.₆₅C₂³⋊₁₆₈C₂, C₂.C₄².₄₅₁C₂², C₂.₅₅(C₂².₂₆C₂⁴), C₂.₁₁₂(C₂³.₃₆C₂³), (C₂×C₄).₆₈₆(C₂×D₄), C₂.₄₈(C₂×C₄⋊D₄), (C₂×C₄⋊D₄).₅₁C₂, (C₂×C₄).₆₇₀(C₄○D₄), (C₂×C₄⋊C₄).₅₅₇C₂², C₂².₅₉₅(C₂×C₄○D₄), (C₂×C₂².D₄)⋊₄₆C₂, C₂.₄₆(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₃₆₄C₂², SmallGroup(128,1586)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₉₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄² — C₂⁴.₅₉₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₉₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₉₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₉₈C₂³

Generators and relations for C₂⁴.₅₉₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=1, f²=cb=bc, g²=b, ab=ba, eae=ac=ca, ad=da, af=fa, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, geg^-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 644 in 344 conjugacy classes, 120 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂²×C₄², C₂×C₄⋊D₄, C₂×C₂².D₄, C₂⁴.₅₉₈C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂⁴.₅₉₈C₂³

Smallest permutation representation of C₂⁴.₅₉₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

kernel

C₂⁴.₅₉₈C₂³

C₂³.₇Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂²×C₄²

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂²×C₄

C₂×C₄

C₂³

# reps

Matrix representation of C₂⁴.₅₉₈C₂³ ►in GL₆(𝔽₅)

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

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	4	0
0	0	0	0	0	4

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

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

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

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

C₂⁴.₅₉₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{598}C_2^3

% in TeX

G:=Group("C2^4.598C2^3");

// GroupNames label

G:=SmallGroup(128,1586);

// by ID

G=gap.SmallGroup(128,1586);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,2019,80]);

// Polycyclic

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

// generators/relations

G = C24.598C23 order 128 = 27

79th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₉₈C₂³ order 128 = 2⁷

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