C2^3.432C2^4 - GroupNames

G = C₂³.₄₃₂C₂⁴ order 128 = 2⁷

149^th central stem extension by C₂³ of C₂⁴

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

Aliases: C₂⁴.₂₅C₂³, C₂³.₄₃₂C₂⁴, C₂².₁₇₁2_-¹⁺⁴, C₂².₂₂₃2₊¹⁺⁴, C₄²⋊₅C₄⋊₁₇C₂, C₄²⋊₄C₄⋊₂₅C₂, C₂³⋊Q₈.₉C₂, (C₂×C₄²).₆₁C₂², (C₂²×C₄).₅₃₃C₂³, C₂³.₁₁D₄.₁₈C₂, (C₂²×Q₈).₁₂₇C₂², C₂³.₈₁C₂³⋊₃₄C₂, C₂³.₆₇C₂³⋊₅₇C₂, C₂³.₆₃C₂³⋊₈₃C₂, C₂³.₈₃C₂³⋊₃₆C₂, C₂⁴.C₂².₃₀C₂, C₂.₄₄(C₂².₄₅C₂⁴), C₂.C₄².₅₄₅C₂², C₂.₅₈(C₂².₄₆C₂⁴), C₂.₂₃(C₂².₄₉C₂⁴), C₂.₃₄(C₂².₅₀C₂⁴), C₂.₄₃(C₂².₃₆C₂⁴), C₂.₇₅(C₂³.₃₆C₂³), (C₄×C₄⋊C₄)⋊₈₄C₂, (C₂×C₄).₁₄₅(C₄○D₄), (C₂×C₄⋊C₄).₂₉₄C₂², C₂².₃₀₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₂C₂², SmallGroup(128,1264)

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

Derived series

C₁ — C₂³ — C₂³.₄₃₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₄×C₄⋊C₄ — C₂³.₄₃₂C₂⁴

Lower central

C₁ — C₂³ — C₂³.₄₃₂C₂⁴

Upper central

C₁ — C₂³ — C₂³.₄₃₂C₂⁴

Jennings

C₁ — C₂³ — C₂³.₄₃₂C₂⁴

Generators and relations for C₂³.₄₃₂C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=c, e²=b, f²=a, g²=ba=ab, ac=ca, ede^-1=gdg^-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 372 in 203 conjugacy classes, 92 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²×Q₈, C₄²⋊₄C₄, C₄×C₄⋊C₄, C₄²⋊₅C₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₇C₂³, C₂³⋊Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₄₃₂C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂³.₃₆C₂³, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₉C₂⁴, C₂².₅₀C₂⁴, C₂³.₄₃₂C₂⁴

Smallest permutation representation of C₂³.₄₃₂C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂³.₄₃₂C₂⁴

C₄²⋊₄C₄

C₄×C₄⋊C₄

C₄²⋊₅C₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₇C₂³

C₂³⋊Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₈₃C₂³

C₂×C₄

C₂²

# reps

Matrix representation of C₂³.₄₃₂C₂⁴ ►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	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	4	0	0	0	0
0	1	0	0	0	0
0	0	2	1	0	0
0	0	2	3	0	0
0	0	0	0	2	2
0	0	0	0	0	3

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

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

4	4	0	0	0	0
2	1	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))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,2,2,0,0,0,0,1,3,0,0,0,0,0,0,2,0,0,0,0,0,2,3],[2,1,0,0,0,0,2,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,3,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,2,1,0,0,0,0,0,0,1,3,0,0,0,0,0,4],[4,2,0,0,0,0,4,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C₂³.₄₃₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{432}C_2^4

% in TeX

G:=Group("C2^3.432C2^4");

// GroupNames label

G:=SmallGroup(128,1264);

// by ID

G=gap.SmallGroup(128,1264);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,675,192]);

// Polycyclic

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

// generators/relations

G = C23.432C24 order 128 = 27

149th central stem extension by C23 of C24

G = C₂³.₄₃₂C₂⁴ order 128 = 2⁷

149^th central stem extension by C₂³ of C₂⁴