C2^4.341C2^3 - GroupNames

G = C₂⁴.₃₄₁C₂³ order 128 = 2⁷

181^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₃₄₁C₂³, C₂³.₄₇₇C₂⁴, C₂².₂₆₀2₊¹⁺⁴, C₂².₁₉₅2_-¹⁺⁴, C₄²⋊₅C₄⋊₂₀C₂, C₂³.₅₆(C₄○D₄), (C₂×C₄²).₅₇₂C₂², (C₂²×C₄).₁₀₈C₂³, (C₂³×C₄).₄₀₈C₂², C₂³.₈Q₈.₃₅C₂, C₂³.₇Q₈.₅₅C₂, C₂³.₁₁D₄.₂₁C₂, C₂³.₈₁C₂³⋊₄₄C₂, C₂³.₆₃C₂³⋊₉₄C₂, C₂³.₈₃C₂³⋊₄₅C₂, C₂⁴.C₂².₃₅C₂, C₂.₅₈(C₂².₄₅C₂⁴), C₂.C₄².₂₁₂C₂², C₂.₃₀(C₂².₃₃C₂⁴), C₂.₈₈(C₂³.₃₆C₂³), C₂.₆₉(C₂².₄₆C₂⁴), C₂.₄₄(C₂².₅₀C₂⁴), C₂.₆₃(C₂².₄₇C₂⁴), (C₄×C₄⋊C₄)⋊₁₀₂C₂, (C₄×C₂²⋊C₄).₆₇C₂, (C₂×C₄).₃₉₅(C₄○D₄), (C₂×C₄⋊C₄).₃₂₄C₂², C₂².₃₅₃(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₉₂C₂², SmallGroup(128,1309)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₄₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₄₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₄₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₄₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₄₁C₂³

Generators and relations for C₂⁴.₃₄₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=c, e²=cb=bc, g²=ba=ab, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 372 in 206 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂⁴.₃₄₁C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₃C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₀C₂⁴, C₂⁴.₃₄₁C₂³

Smallest permutation representation of C₂⁴.₃₄₁C₂³
►On 64 points

Generators in S₆₄

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

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

(2 52)(4 50)(5 34)(6 8)(7 36)(10 22)(12 24)(14 26)(16 28)(17 19)(18 32)(20 30)(29 31)(33 35)(37 39)(38 64)(40 62)(42 54)(44 56)(45 47)(46 60)(48 58)(57 59)(61 63)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4X	4Y	4Z	4AA	4AB
order	1	2	···	2	2	2	4	···	4	4	···	4	4	4	4	4
size	1	1	···	1	4	4	2	···	2	4	···	4	8	8	8	8

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₄₁C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₃₄₁C₂³ ►in GL₆(𝔽₅)

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

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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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

1	0	0	0	0	0
0	4	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	2	4

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

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

C₂⁴.₃₄₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{341}C_2^3

% in TeX

G:=Group("C2^4.341C2^3");

// GroupNames label

G:=SmallGroup(128,1309);

// by ID

G=gap.SmallGroup(128,1309);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,723,675,192]);

// Polycyclic

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

// generators/relations

G = C24.341C23 order 128 = 27

181st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₄₁C₂³ order 128 = 2⁷

181^st non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²