C2^4.286C2^3 - GroupNames

G = C₂⁴.₂₈₆C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₈₆C₂³, C₂³.₃₆₆C₂⁴, C₂².₁₇₂2₊¹⁺⁴, C₂².₁₂₇2_-¹⁺⁴, C₂²⋊C₄.₁₂₉D₄, C₂³.₁₇₄(C₂×D₄), C₂.₄₉(D₄⋊₅D₄), C₂.₃₁(D₄⋊₆D₄), C₂³.₃₄(C₄○D₄), (C₂³×C₄).₉₀C₂², C₂³.Q₈⋊₁₇C₂, C₂³.₈Q₈⋊₅₂C₂, C₂³.₁₁D₄⋊₁₆C₂, C₂³.₃₄D₄⋊₂₇C₂, (C₂²×C₄).₈₁₉C₂³, (C₂×C₄²).₅₀₉C₂², C₂³.₁₀D₄.₆C₂, C₂².₂₄₆(C₂²×D₄), C₂⁴.C₂²⋊₄₉C₂, (C₂²×D₄).₅₂₀C₂², C₂³.₆₃C₂³⋊₄₅C₂, C₂³.₆₅C₂³⋊₆₀C₂, C₂.₃₈(C₂².₁₉C₂⁴), C₂.C₄².₁₂₃C₂², C₂.₁₇(C₂².₃₃C₂⁴), C₂.₁₆(C₂².₅₀C₂⁴), C₂.₂₄(C₂².₄₇C₂⁴), C₂.₃₄(C₂³.₃₆C₂³), (C₂×C₄×D₄).₅₃C₂, (C₄×C₂²⋊C₄)⋊₆₅C₂, (C₂×C₄).₈₉₇(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₁C₂, (C₂×C₄).₁₁₆(C₄○D₄), (C₂×C₄⋊C₄).₂₄₆C₂², C₂².₂₄₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₄₀C₂², SmallGroup(128,1198)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₈₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₂²⋊C₄ — C₂⁴.₂₈₆C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₈₆C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₈₆C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₈₆C₂³

Generators and relations for C₂⁴.₂₈₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=1, f²=b, g²=cb=bc, eae=gag^-1=ab=ba, faf^-1=ac=ca, ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef^-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 500 in 266 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂×C₄×D₄, C₂×C₄²⋊₂C₂, C₂⁴.₂₈₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂⁴.₂₈₆C₂³

Smallest permutation representation of C₂⁴.₂₈₆C₂³
►On 64 points

Generators in S₆₄

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

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

(2 16)(4 14)(5 55)(6 8)(7 53)(10 36)(12 34)(17 45)(18 20)(19 47)(21 41)(22 24)(23 43)(26 52)(28 50)(30 40)(32 38)(42 44)(46 48)(54 56)(57 62)(58 60)(59 64)(61 63)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₂₈₆C₂³

C₄×C₂²⋊C₄

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂×C₄×D₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₂₈₆C₂³ ►in GL₆(𝔽₅)

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	2	2
0	0	0	0	1	3

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

0	1	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	3	0
0	0	0	0	0	3

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

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

C₂⁴.₂₈₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{286}C_2^3

% in TeX

G:=Group("C2^4.286C2^3");

// GroupNames label

G:=SmallGroup(128,1198);

// by ID

G=gap.SmallGroup(128,1198);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100,675,192]);

// Polycyclic

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

// generators/relations

G = C24.286C23 order 128 = 27

126th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₈₆C₂³ order 128 = 2⁷

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