C2^4.340C2^3 - GroupNames

G = C₂⁴.₃₄₀C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₄₀C₂³, C₂³.₄₇₆C₂⁴, C₂².₂₅₉2₊¹⁺⁴, C₄²⋊₈C₄⋊₄₇C₂, C₂³.₅₅(C₄○D₄), (C₂×C₄²).₇₁C₂², C₂³.Q₈⋊₃₂C₂, C₂³.₈Q₈⋊₇₀C₂, C₂³.₃₄D₄⋊₃₈C₂, C₂³.₁₁D₄⋊₄₇C₂, (C₂²×C₄).₁₀₇C₂³, (C₂³×C₄).₁₂₂C₂², C₂⁴.C₂²⋊₉₀C₂, C₂³.₈₄C₂³⋊₅C₂, C₂³.₁₀D₄.₂₄C₂, C₂³.₂₃D₄.₃₈C₂, (C₂²×D₄).₁₇₆C₂², C₂³.₆₃C₂³⋊₉₃C₂, C₂.₂₈(C₂².₃₂C₂⁴), C₂.₅₇(C₂².₄₅C₂⁴), C₂.C₄².₄₉₁C₂², C₂.₆₂(C₂².₄₇C₂⁴), C₂.₈₇(C₂³.₃₆C₂³), (C₄×C₂²⋊C₄)⋊₁₆C₂, (C₂×C₄).₁₅₄(C₄○D₄), (C₂×C₄⋊C₄).₃₂₃C₂², C₂².₃₅₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₁C₂², SmallGroup(128,1308)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₄₀C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₄₀C₂³

Generators and relations for C₂⁴.₃₄₀C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=bcd, g²=cb=bc, eae^-1=gag^-1=ab=ba, ac=ca, ad=da, faf=acd, bd=db, geg^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, gfg^-1=cdf >

Subgroups: 468 in 234 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₄C₂³, C₂⁴.₃₄₀C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₂C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂⁴.₃₄₀C₂³

Smallest permutation representation of C₂⁴.₃₄₀C₂³
►On 64 points

Generators in S₆₄

(2 12)(4 10)(5 7)(6 39)(8 37)(13 43)(15 41)(17 19)(18 46)(20 48)(21 23)(22 50)(24 52)(26 56)(28 54)(30 60)(32 58)(33 61)(34 36)(35 63)(38 40)(45 47)(49 51)(62 64)

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

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

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

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

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

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

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

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

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₃₄₀C₂³

C₄×C₂²⋊C₄

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₄₀C₂³ ►in GL₆(𝔽₅)

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

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

0	2	0	0	0	0
3	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	2

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

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

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

C₂⁴.₃₄₀C₂³ in GAP, Magma, Sage, TeX

C_2^4._{340}C_2^3

% in TeX

G:=Group("C2^4.340C2^3");

// GroupNames label

G:=SmallGroup(128,1308);

// by ID

G=gap.SmallGroup(128,1308);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,792,758,723,675,136]);

// Polycyclic

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

// generators/relations

G = C24.340C23 order 128 = 27

180th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₄₀C₂³ order 128 = 2⁷

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