C2^4.311C2^3 - GroupNames

G = C₂⁴.₃₁₁C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₃₁₁C₂³, C₂³.₄₂₁C₂⁴, C₂².₂₁₄2₊¹⁺⁴, C₂².₁₆₃2_-¹⁺⁴, C₂²⋊C₄.₇₄D₄, C₂³.₄₄(C₂×D₄), C₂.₄₈(D₄⋊₆D₄), C₂.₆₄(D₄⋊₅D₄), C₂³.₄Q₈⋊₁₈C₂, C₂³.₇Q₈⋊₆₀C₂, C₂³.₁₁D₄⋊₃₇C₂, (C₂×C₄²).₅₃₆C₂², (C₂³×C₄).₃₈₅C₂², (C₂²×C₄).₅₃₁C₂³, C₂².₂₈₄(C₂²×D₄), C₂⁴.C₂²⋊₇₃C₂, C₂³.₁₀D₄.₁₅C₂, C₂³.₂₃D₄.₃₁C₂, (C₂²×D₄).₁₅₇C₂², (C₂²×Q₈).₁₂₄C₂², C₂³.₇₈C₂³⋊₁₄C₂, C₂³.₆₃C₂³⋊₇₄C₂, C₂³.₆₇C₂³⋊₅₄C₂, C₂.₃₇(C₂².₄₅C₂⁴), C₂.C₄².₁₆₉C₂², C₂.₂₉(C₂².₂₆C₂⁴), C₂.₂₈(C₂².₅₀C₂⁴), C₂.₃₈(C₂².₃₆C₂⁴), C₂.₆₄(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₈₀C₂, (C₂×C₄).₆₉(C₂×D₄), (C₄×C₂²⋊C₄)⋊₈₀C₂, (C₂×C₄²⋊₂C₂)⋊₈C₂, (C₂×C₄).₁₄₀(C₄○D₄), (C₂×C₄⋊C₄).₂₈₄C₂², (C₂×C₄.₄D₄).₂₃C₂, C₂².₂₉₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₆₆C₂², SmallGroup(128,1253)

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

Derived series

C₁ — C₂³ — C₂⁴.₃₁₁C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₃₁₁C₂³

Lower central

C₁ — C₂³ — C₂⁴.₃₁₁C₂³

Upper central

C₁ — C₂³ — C₂⁴.₃₁₁C₂³

Jennings

C₁ — C₂³ — C₂⁴.₃₁₁C₂³

Generators and relations for C₂⁴.₃₁₁C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=f²=1, d²=c, e²=a, g²=ba=ab, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 500 in 259 conjugacy classes, 100 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄.₄D₄, C₄²⋊₂C₂, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³.₁₀D₄, C₂³.₇₈C₂³, C₂³.₁₁D₄, C₂³.₄Q₈, C₂×C₄.₄D₄, C₂×C₄²⋊₂C₂, C₂⁴.₃₁₁C₂³
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, C₂².₃₆C₂⁴, D₄⋊₅D₄, D₄⋊₆D₄, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂⁴.₃₁₁C₂³

Smallest permutation representation of C₂⁴.₃₁₁C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₃₁₁C₂³

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³.₁₀D₄

C₂³.₇₈C₂³

C₂³.₁₁D₄

C₂³.₄Q₈

C₂×C₄.₄D₄

C₂×C₄²⋊₂C₂

C₂²⋊C₄

C₂×C₄

C₂²

# reps

Matrix representation of C₂⁴.₃₁₁C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

C₂⁴.₃₁₁C₂³ in GAP, Magma, Sage, TeX

C_2^4._{311}C_2^3

% in TeX

G:=Group("C2^4.311C2^3");

// GroupNames label

G:=SmallGroup(128,1253);

// by ID

G=gap.SmallGroup(128,1253);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,675,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=a,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,b*c=c*b,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 = C24.311C23 order 128 = 27

151st non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₃₁₁C₂³ order 128 = 2⁷

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