C2^4.271C2^3 - GroupNames

G = C₂⁴.₂₇₁C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₂₇₁C₂³, C₂³.₃₄₇C₂⁴, C₂².₁₁₄2_-¹⁺⁴, C₂².₁₅₅2₊¹⁺⁴, (C₂×D₄).₂₈₉D₄, C₂³.₁₆₇(C₂×D₄), C₂.₂₅(D₄⋊₆D₄), C₂.₃₈(D₄⋊₅D₄), (C₂³×C₄).₇₈C₂², C₂³.₈Q₈⋊₄₀C₂, C₂³.₃₄D₄⋊₂₄C₂, C₂³.₁₁D₄⋊₁₃C₂, (C₂²×C₄).₅₁₃C₂³, (C₂×C₄²).₄₉₀C₂², C₂².₂₂₇(C₂²×D₄), C₄.₈₀(C₂².D₄), (C₂²×D₄).₅₀₉C₂², (C₂²×Q₈).₁₀₄C₂², C₂³.₆₇C₂³⋊₄₁C₂, C₂³.₆₅C₂³⋊₅₃C₂, C₂⁴.₃C₂².₃₂C₂, C₂.C₄².₁₀₄C₂², C₂.₇(C₂².₅₃C₂⁴), C₂.₁₆(C₂².₃₆C₂⁴), C₂.₁₃(C₂².₅₀C₂⁴), C₂.₂₅(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₅₆C₂, (C₂×C₄×D₄).₄₇C₂, (C₂×C₄).₁₀₇₈(C₂×D₄), (C₂×C₄).₁₀₄(C₄○D₄), (C₂×C₄⋊C₄).₂₂₉C₂², (C₂×C₄.₄D₄).₂₀C₂, C₂².₂₂₄(C₂×C₄○D₄), C₂.₂₅(C₂×C₂².D₄), (C₂×C₂²⋊C₄).₄₅₄C₂², SmallGroup(128,1179)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₇₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂⁴.₂₇₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₂₇₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₂₇₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₂₇₁C₂³

Generators and relations for C₂⁴.₂₇₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=c, f²=cb=bc, g²=b, eae^-1=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: 516 in 270 conjugacy classes, 104 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₄.₄D₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₄×C₄⋊C₄, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₆₇C₂³, C₂³.₁₁D₄, C₂×C₄×D₄, C₂×C₄.₄D₄, C₂⁴.₂₇₁C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂×C₂².D₄, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₅₀C₂⁴, C₂².₅₃C₂⁴, C₂⁴.₂₇₁C₂³

Smallest permutation representation of C₂⁴.₂₇₁C₂³
►On 64 points

Generators in S₆₄

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

(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

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

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

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

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

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

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

C₂⁴.₂₇₁C₂³

C₄×C₄⋊C₄

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₆₇C₂³

C₂³.₁₁D₄

C₂×C₄×D₄

C₂×C₄.₄D₄

C₂×D₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₂₇₁C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

C₂⁴.₂₇₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{271}C_2^3

% in TeX

G:=Group("C2^4.271C2^3");

// GroupNames label

G:=SmallGroup(128,1179);

// by ID

G=gap.SmallGroup(128,1179);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,268,675,80]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=c*b=b*c,g^2=b,e*a*e^-1=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 = C24.271C23 order 128 = 27

111st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₂₇₁C₂³ order 128 = 2⁷

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