C2^4.435C2^3 - GroupNames

G = C₂⁴.₄₃₅C₂³ order 128 = 2⁷

275^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₂⁴.₄₃₅C₂³, C₂³.₆₅₀C₂⁴, C₂².₄₂₃2₊¹⁺⁴, C₂².₃₂₀2_-¹⁺⁴, C₂³.₉₁(C₄○D₄), (C₂×C₄²).₉₆C₂², C₂³.Q₈⋊₈₀C₂, C₂³.₃₄D₄⋊₅₆C₂, (C₂²×C₄).₅₇₁C₂³, (C₂³×C₄).₄₈₈C₂², C₂³.₈Q₈⋊₁₂₇C₂, C₂³.₇Q₈⋊₁₀₅C₂, C₂³.₁₁D₄⋊₁₀₉C₂, C₂³.₁₀D₄.₅₇C₂, C₂³.₂₃D₄.₆₆C₂, (C₂²×D₄).₂₆₇C₂², C₂⁴.C₂²⋊₁₅₈C₂, C₂.₈₃(C₂².₃₂C₂⁴), C₂³.₆₃C₂³⋊₁₆₅C₂, C₂³.₈₃C₂³⋊₁₀₁C₂, C₂.C₄².₃₅₄C₂², C₂.₁₀₂(C₂².₄₅C₂⁴), C₂.₉₀(C₂².₃₃C₂⁴), C₂.₃₁(C₂².₅₆C₂⁴), C₂.₉₆(C₂².₄₇C₂⁴), C₂.₉₆(C₂².₄₆C₂⁴), (C₂×C₄).₄₅₁(C₄○D₄), (C₂×C₄⋊C₄).₄₆₁C₂², C₂².₅₁₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₅C₂², SmallGroup(128,1482)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₃₅C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₄₃₅C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₃₅C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₃₅C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₃₅C₂³

Generators and relations for C₂⁴.₄₃₅C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=db=bd, g²=cb=bc, faf=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, fef=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.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂⁴.₄₃₅C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₃₅C₂³

Smallest permutation representation of C₂⁴.₄₃₅C₂³
►On 64 points

Generators in S₆₄

(2 56)(4 54)(5 63)(6 49)(7 61)(8 51)(9 23)(10 35)(11 21)(12 33)(14 27)(16 25)(17 32)(19 30)(22 38)(24 40)(34 39)(36 37)(41 64)(42 50)(43 62)(44 52)(46 59)(48 57)

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

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

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

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

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

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

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

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

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

C₂⁴.₄₃₅C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₃₅C₂³ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	4	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	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

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

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

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

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

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

C₂⁴.₄₃₅C₂³ in GAP, Magma, Sage, TeX

C_2^4._{435}C_2^3

% in TeX

G:=Group("C2^4.435C2^3");

// GroupNames label

G:=SmallGroup(128,1482);

// by ID

G=gap.SmallGroup(128,1482);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,268,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d*b=b*d,g^2=c*b=b*c,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,f*e*f=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 = C24.435C23 order 128 = 27

275th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₃₅C₂³ order 128 = 2⁷

275^th non-split extension by C₂⁴ of C₂³ acting via C₂³/C₂=C₂²