C2^4.435C2^3

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, C₂³.₇Q₈⋊₁₀₅C₂, C₂³.₈Q₈⋊₁₂₇C₂, C₂³.₁₁D₄⋊₁₀₉C₂, C₂³.₂₃D₄.₆₆C₂, C₂³.₁₀D₄.₅₇C₂, (C₂²×D₄).₂₆₇C₂², C₂⁴.C₂²⋊₁₅₈C₂, C₂.₈₃(C₂².₃₂C₂⁴), C₂³.₈₃C₂³⋊₁₀₁C₂, C₂³.₆₃C₂³⋊₁₆₅C₂, C₂.C₄².₃₅₄C₂², C₂.₁₀₂(C₂².₄₅C₂⁴), C₂.₉₆(C₂².₄₆C₂⁴), C₂.₉₆(C₂².₄₇C₂⁴), C₂.₃₁(C₂².₅₆C₂⁴), C₂.₉₀(C₂².₃₃C₂⁴), (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

Subgroups: 468 in 224 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂^[×7], C₂^[×4], C₄^[×14], C₂²^[×7], C₂²^[×20], C₂×C₄^[×2], C₂×C₄^[×46], D₄^[×4], C₂³, C₂³^[×4], C₂³^[×12], C₄², C₂²⋊C₄^[×14], C₄⋊C₄^[×8], C₂²×C₄^[×13], C₂²×C₄^[×8], C₂×D₄^[×5], C₂⁴^[×2], C₂.C₄²^[×14], C₂×C₄², C₂×C₂²⋊C₄^[×10], C₂×C₄⋊C₄^[×7], C₂³×C₄^[×2], C₂²×D₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂³.₈Q₈^[×2], C₂³.₂₃D₄^[×2], C₂³.₆₃C₂³, C₂⁴.C₂²^[×2], C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₃C₂³^[×3], C₂⁴.₄₃₅C₂³

Quotients:
C₁, C₂^[×15], C₂²^[×35], C₂³^[×15], C₄○D₄^[×6], C₂⁴, C₂×C₄○D₄^[×3], 2₊⁽¹⁺⁴⁾^[×3], 2_-⁽¹⁺⁴⁾, C₂².₃₂C₂⁴^[×2], C₂².₃₃C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₃₅C₂³

Generators and relations
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 >

Smallest permutation representation
►On 64 points

Generators in S₆₄

(2 56)(4 54)(5 37)(6 24)(7 39)(8 22)(9 25)(11 27)(14 30)(16 32)(17 62)(18 47)(19 64)(20 45)(21 43)(23 41)(33 48)(34 61)(35 46)(36 63)(38 42)(40 44)(49 57)(51 59)

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;

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

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

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

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