C2^4.432C2^3 - GroupNames

G = C₂⁴.₄₃₂C₂³ order 128 = 2⁷

272^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₄₃₂C₂³, C₂³.₆₄₆C₂⁴, C₂².₄₁₉2₊¹⁺⁴, C₂².₃₁₈2_-¹⁺⁴, C₂³.₈₈(C₄○D₄), (C₂×C₄²).₉₄C₂², C₂³.₄Q₈⋊₅₈C₂, C₂³.₃₄D₄⋊₅₄C₂, (C₂³×C₄).₁₅₉C₂², (C₂²×C₄).₂₀₅C₂³, C₂³.₈Q₈⋊₁₂₄C₂, C₂³.₇Q₈⋊₁₀₃C₂, C₂³.₁₁D₄⋊₁₀₇C₂, C₂³.₂₃D₄.₆₄C₂, C₂³.₁₀D₄.₅₆C₂, (C₂²×D₄).₂₆₄C₂², C₂⁴.C₂²⋊₁₅₆C₂, C₂³.₈₄C₂³⋊₁₀C₂, C₂³.₈₃C₂³⋊₉₈C₂, C₂³.₈₁C₂³⋊₁₀₉C₂, C₂³.₆₃C₂³⋊₁₆₂C₂, C₂.₉₈(C₂².₄₅C₂⁴), C₂.₂₄(C₂².₅₄C₂⁴), C₂.C₄².₃₅₀C₂², C₂.₈₈(C₂².₃₃C₂⁴), C₂.₉₄(C₂².₃₆C₂⁴), C₂.₉₃(C₂².₄₇C₂⁴), (C₂×C₄).₄₄₇(C₄○D₄), (C₂×C₄⋊C₄).₄₅₇C₂², C₂².₅₀₇(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₆₄C₂², SmallGroup(128,1478)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₃₂C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₄₃₂C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₃₂C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₃₂C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₃₂C₂³

Generators and relations for C₂⁴.₄₃₂C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=bcd, f²=cb=bc, g²=b, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, cd=dc, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 468 in 224 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂³.₈₄C₂³, C₂⁴.₄₃₂C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₃₂C₂³

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

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₃₂C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂³.₈₄C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₃₂C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

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	3
0	0	0	0	0	4

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

C₂⁴.₄₃₂C₂³ in GAP, Magma, Sage, TeX

C_2^4._{432}C_2^3

% in TeX

G:=Group("C2^4.432C2^3");

// GroupNames label

G:=SmallGroup(128,1478);

// by ID

G=gap.SmallGroup(128,1478);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.432C23 order 128 = 27

272nd non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₃₂C₂³ order 128 = 2⁷

272^nd non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²