C2^3.165C2^4 - GroupNames

G = C₂³.₁₆₅C₂⁴ order 128 = 2⁷

18^th central extension by C₂³ of C₂⁴

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

Aliases: C₂³.₁₆₅C₂⁴, C₂⁴.₅₂₉C₂³, (C₂×C₄²)⋊₂₁C₄, C₄ ○(C₄²⋊₈C₄), C₄ ○(C₄²⋊₅C₄), C₄²⋊₄C₄⋊₁C₂, C₄²⋊₈C₄⋊₇₆C₂, C₄²⋊₅C₄⋊₄₀C₂, C₄².₂₉₈(C₂×C₄), (C₂²×C₄²).₁₉C₂, C₂².₅₆(C₂³×C₄), C₄ ○₂(C₂³.₇Q₈), C₄.₆₃(C₄²⋊C₂), C₄ ○₂(C₂³.₃₄D₄), C₂³.₂₁₄(C₄○D₄), C₂³.₂₀₈(C₂²×C₄), (C₂²×C₄).₄₄₃C₂³, (C₂³×C₄).₆₄₅C₂², C₂³.₇Q₈.₈₁C₂, (C₂×C₄²).₁₀₀₁C₂², C₂³.₃₄D₄.₃₈C₂, C₂².₃₄(C₄²⋊C₂), C₂.C₄².₄₆₄C₂², C₂.₁(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₁₃C₂, (C₄×C₂²⋊C₄).₄C₂, (C₂×C₄)○(C₄²⋊₅C₄), (C₂×C₄)○(C₄²⋊₈C₄), C₂².₅₈(C₂×C₄○D₄), (C₂×C₄).₆₃₃(C₄○D₄), (C₂×C₄⋊C₄).₇₈₄C₂², (C₂²×C₄).₄₅₃(C₂×C₄), (C₂×C₄).₄₈₇(C₂²×C₄), C₂.₁₁(C₂×C₄²⋊C₂), (C₂×C₄)○(C₂³.₃₄D₄), (C₂×C₂²⋊C₄).₄₁₂C₂², SmallGroup(128,1015)

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

Derived series

C₁ — C₂² — C₂³.₁₆₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄² — C₂³.₁₆₅C₂⁴

Lower central

C₁ — C₂² — C₂³.₁₆₅C₂⁴

Upper central

C₁ — C₂²×C₄ — C₂³.₁₆₅C₂⁴

Jennings

C₁ — C₂³ — C₂³.₁₆₅C₂⁴

Generators and relations for C₂³.₁₆₅C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=d, f²=c, g²=b, ab=ba, eae^-1=ac=ca, ad=da, af=fa, ag=ga, bc=cb, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 428 in 284 conjugacy classes, 156 normal (22 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄²⋊₄C₄, C₄×C₂²⋊C₄, C₄×C₄⋊C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₄²⋊₈C₄, C₄²⋊₅C₄, C₂²×C₄², C₂³.₁₆₅C₂⁴
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₄○D₄, C₂⁴, C₄²⋊C₂, C₂³×C₄, C₂×C₄○D₄, C₂×C₄²⋊C₂, C₂³.₃₆C₂³, C₂³.₁₆₅C₂⁴

Smallest permutation representation of C₂³.₁₆₅C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4AB	4AC	···	4AR
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	4	···	4
size	1	1	···	1	2	2	2	2	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim

type

image

C₁

C₂

C₄

C₄○D₄

kernel

C₂³.₁₆₅C₂⁴

C₄²⋊₄C₄

C₄×C₂²⋊C₄

C₄×C₄⋊C₄

C₂³.₇Q₈

C₂³.₃₄D₄

C₄²⋊₈C₄

C₄²⋊₅C₄

C₂²×C₄²

C₂×C₄²

C₂×C₄

C₂³

# reps

Matrix representation of C₂³.₁₆₅C₂⁴ ►in GL₆(𝔽₅)

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	4	0
0	0	0	0	3	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	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
0	4	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

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

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

3	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	3	0
0	0	0	0	0	3

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

C₂³.₁₆₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{165}C_2^4

% in TeX

G:=Group("C2^3.165C2^4");

// GroupNames label

G:=SmallGroup(128,1015);

// by ID

G=gap.SmallGroup(128,1015);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,80]);

// Polycyclic

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

// generators/relations

G = C23.165C24 order 128 = 27

18th central extension by C23 of C24

G = C₂³.₁₆₅C₂⁴ order 128 = 2⁷

18^th central extension by C₂³ of C₂⁴