C2^4.579C2^3 - GroupNames

G = C₂⁴.₅₇₉C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₇₉C₂³, C₂³.₄₀₃C₂⁴, C₂².₁₅₂2_-¹⁺⁴, C₄²⋊₈C₄⋊₃₁C₂, (C₂²×C₄).₃₉₀D₄, C₂³.₆₁₇(C₂×D₄), C₂³.₃₁₁(C₄○D₄), (C₂×C₄²).₅₂₃C₂², (C₂²×C₄).₅₂₄C₂³, (C₂³×C₄).₁₀₀C₂², C₂².₂₇₉(C₂²×D₄), C₂³.₃₄D₄.₁₆C₂, C₄.₅₃(C₂².D₄), C₂².₂₉(C₄.₄D₄), (C₂²×Q₈).₁₂₀C₂², C₂³.₈₃C₂³⋊₂₆C₂, C₂³.₆₇C₂³⋊₅₃C₂, C₂.C₄².₁₅₄C₂², C₂.₄₃(C₂².₄₆C₂⁴), C₂.₁₉(C₂³.₃₈C₂³), (C₂×C₄).₈₃₂(C₂×D₄), (C₂²×C₄⋊C₄).₃₈C₂, (C₄×C₂²⋊C₄).₅₃C₂, C₂.₁₉(C₂×C₄.₄D₄), (C₂×C₂²⋊Q₈).₃₃C₂, (C₂×C₄).₈₁₅(C₄○D₄), (C₂×C₄⋊C₄).₂₇₀C₂², C₂².₂₈₀(C₂×C₄○D₄), C₂.₃₈(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₆₅C₂², SmallGroup(128,1235)

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

Derived series

C₁ — C₂³ — C₂⁴.₅₇₉C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₅₇₉C₂³

Lower central

C₁ — C₂³ — C₂⁴.₅₇₉C₂³

Upper central

C₁ — C₂³ — C₂⁴.₅₇₉C₂³

Jennings

C₁ — C₂³ — C₂⁴.₅₇₉C₂³

Generators and relations for C₂⁴.₅₇₉C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=cb=bc, g²=b, ab=ba, ac=ca, faf^-1=ad=da, ae=ea, ag=ga, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 452 in 256 conjugacy classes, 108 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₆₇C₂³, C₂³.₈₃C₂³, C₂²×C₄⋊C₄, C₂×C₂²⋊Q₈, C₂⁴.₅₇₉C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×C₂².D₄, C₂×C₄.₄D₄, C₂³.₃₈C₂³, C₂².₄₆C₂⁴, C₂⁴.₅₇₉C₂³

Smallest permutation representation of C₂⁴.₅₇₉C₂³
►On 64 points

Generators in S₆₄

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

(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4V	4W	4X	4Y	4Z
order	1	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	4	4	4
size	1	1	···	1	2	2	2	2	2	2	2	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2_-¹⁺⁴

kernel

C₂⁴.₅₇₉C₂³

C₄×C₂²⋊C₄

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₆₇C₂³

C₂³.₈₃C₂³

C₂²×C₄⋊C₄

C₂×C₂²⋊Q₈

C₂²×C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₅₇₉C₂³ ►in GL₆(𝔽₅)

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	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

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	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

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

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

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	3
0	0	0	0	1	1

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

C₂⁴.₅₇₉C₂³ in GAP, Magma, Sage, TeX

C_2^4._{579}C_2^3

% in TeX

G:=Group("C2^4.579C2^3");

// GroupNames label

G:=SmallGroup(128,1235);

// by ID

G=gap.SmallGroup(128,1235);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,268,675,80]);

// Polycyclic

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

// generators/relations

G = C24.579C23 order 128 = 27

60th non-split extension by C24 of C23 acting via C23/C22=C2

G = C₂⁴.₅₇₉C₂³ order 128 = 2⁷

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