C2^4.426C2^3 - GroupNames

G = C₂⁴.₄₂₆C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₂₆C₂³, C₂³.₆₃₈C₂⁴, C₂².₄₁₁2₊¹⁺⁴, C₂².₃₁₀2_-¹⁺⁴, C₄²⋊₅C₄⋊₂₈C₂, (C₂×C₄²).₈₈C₂², C₂³.₁₈₂(C₄○D₄), (C₂³×C₄).₄₈₀C₂², (C₂²×C₄).₅₆₄C₂³, C₂³.₄Q₈.₂₂C₂, C₂³.₇Q₈.₆₇C₂, C₂³.₈Q₈.₅₂C₂, C₂³.₁₁D₄.₄₀C₂, C₂³.₃₄D₄.₂₈C₂, C₂³.₈₃C₂³⋊₉₁C₂, C₂⁴.C₂².₅₇C₂, C₂³.₈₁C₂³⋊₁₀₄C₂, C₂³.₆₃C₂³⋊₁₅₄C₂, C₂³.₆₅C₂³⋊₁₃₉C₂, C₂.₉₀(C₂².₄₅C₂⁴), C₂.C₄².₃₄₂C₂², C₂.₄₈(C₂².₃₄C₂⁴), C₂.₈₂(C₂².₃₃C₂⁴), C₂.₈₉(C₂².₄₆C₂⁴), C₂.₈₉(C₂².₃₆C₂⁴), C₂.₃₃(C₂².₅₇C₂⁴), (C₂×C₄).₄₄₀(C₄○D₄), (C₂×C₄⋊C₄).₄₄₉C₂², C₂².₄₉₉(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₅₉C₂², SmallGroup(128,1470)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₂₆C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₃₄D₄ — C₂⁴.₄₂₆C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₂₆C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₂₆C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₂₆C₂³

Generators and relations for C₂⁴.₄₂₆C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=g²=1, d²=ca=ac, e²=b, f²=a, ab=ba, ede^-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 372 in 196 conjugacy classes, 88 normal (82 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂³.₄Q₈, C₂³.₈₃C₂³, C₂⁴.₄₂₆C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₅₇C₂⁴, C₂⁴.₄₂₆C₂³

Smallest permutation representation of C₂⁴.₄₂₆C₂³
►On 64 points

Generators in S₆₄

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

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

(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)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4P	4Q	···	4V
order	1	2	···	2	2	2	4	···	4	4	···	4
size	1	1	···	1	4	4	4	···	4	8	···	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₂₆C₂³

C₂³.₇Q₈

C₂³.₃₄D₄

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂³.₄Q₈

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₂₆C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

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

C₂⁴.₄₂₆C₂³ in GAP, Magma, Sage, TeX

C_2^4._{426}C_2^3

% in TeX

G:=Group("C2^4.426C2^3");

// GroupNames label

G:=SmallGroup(128,1470);

// by ID

G=gap.SmallGroup(128,1470);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,100,1571,346]);

// Polycyclic

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

// generators/relations

G = C24.426C23 order 128 = 27

266th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₂₆C₂³ order 128 = 2⁷

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