C2^4.326C2^3 - GroupNames

G = C₂⁴.₃₂₆C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₂₆C₂³, C₂³.₄₅₃C₂⁴, C₂².₂₃₈2₊¹⁺⁴, C₂².₁₈₄2_-¹⁺⁴, C₂²⋊C₄.₇₆D₄, C₂³.₅₂(C₂×D₄), C₄²⋊₈C₄⋊₄₂C₂, C₂.₅₃(D₄⋊₆D₄), C₂.₇₃(D₄⋊₅D₄), C₂³.₇Q₈⋊₆₈C₂, C₂³.₄Q₈⋊₂₂C₂, C₂³.₈Q₈⋊₆₅C₂, C₂³.₁₅₂(C₄○D₄), C₂³.₁₁D₄⋊₄₃C₂, (C₂²×C₄).₅₃₉C₂³, (C₂×C₄²).₅₅₈C₂², (C₂³×C₄).₃₉₈C₂², C₂².₃₀₄(C₂²×D₄), C₂⁴.C₂²⋊₈₀C₂, C₂³.₂₃D₄.₃₄C₂, C₂³.₁₀D₄.₁₉C₂, (C₂²×D₄).₁₆₈C₂², C₂³.₆₅C₂³⋊₈₆C₂, C₂³.₈₁C₂³⋊₃₈C₂, C₂³.₆₃C₂³⋊₈₄C₂, C₂.₄₆(C₂².₄₅C₂⁴), C₂.C₄².₁₉₀C₂², C₂.₃₃(C₂².₂₆C₂⁴), C₂.₆₁(C₂².₄₆C₂⁴), C₂.₂₄(C₂².₃₃C₂⁴), C₂.₇₇(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄)⋊₈₄C₂, (C₂×C₄).₉₀₆(C₂×D₄), (C₂×C₄²⋊₂C₂)⋊₉C₂, (C₂×C₄).₃₈₆(C₄○D₄), (C₂×C₄⋊C₄).₃₀₆C₂², C₂².₃₂₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₀₅C₂², (C₂×C₂².D₄).₁₇C₂, SmallGroup(128,1285)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₂₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₂₆C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₂₆C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₂₆C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₂₆C₂³

Generators and relations for C₂⁴.₃₂₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=1, f²=db=bd, g²=c, ab=ba, eae=ac=ca, faf^-1=ad=da, ag=ga, bc=cb, fef^-1=geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 500 in 262 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, C₂².D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₇Q₈, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂×C₂².D₄, C₂×C₄²⋊₂C₂, C₂⁴.₃₂₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₂₆C₂⁴, C₂².₃₃C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂⁴.₃₂₆C₂³

Smallest permutation representation of C₂⁴.₃₂₆C₂³
►On 64 points

Generators in S₆₄

(2 27)(4 25)(5 53)(6 57)(7 55)(8 59)(10 14)(12 16)(17 43)(18 45)(19 41)(20 47)(21 44)(22 46)(23 42)(24 48)(30 51)(32 49)(34 39)(36 37)(54 62)(56 64)(58 63)(60 61)

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

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

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

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

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

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

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

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

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

C₂⁴.₃₂₆C₂³

C₄×C₂²⋊C₄

C₂³.₇Q₈

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂×C₂².D₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₂₆C₂³ ►in GL₆(𝔽₅)

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	1	0
0	0	0	0	1	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

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

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

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

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

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

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

C₂⁴.₃₂₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{326}C_2^3

% in TeX

G:=Group("C2^4.326C2^3");

// GroupNames label

G:=SmallGroup(128,1285);

// by ID

G=gap.SmallGroup(128,1285);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,136]);

// Polycyclic

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

// generators/relations

G = C24.326C23 order 128 = 27

166th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₂₆C₂³ order 128 = 2⁷

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