C2xC2^2.47C2^4 - GroupNames

G = C₂×C₂².₄₇C₂⁴ order 128 = 2⁷

Direct product of C₂ and C₂².₄₇C₂⁴

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

Aliases: C₂×C₂².₄₇C₂⁴, C₂³.₂₇C₂⁴, C₂².₆₀C₂⁵, C₄².₅₅₈C₂³, C₂⁴.₆₁₇C₂³, C₂².₁₁₂2₊¹⁺⁴, (C₄×D₄)⋊₁₀₇C₂², C₄⋊D₄⋊₇₄C₂², C₄⋊C₄.₂₈₉C₂³, (C₂×C₄).₁₆₅C₂⁴, (C₂×D₄).₄₅₆C₂³, C₂²⋊C₄.₈₄C₂³, C₄².C₂⋊₄₈C₂², C₄²⋊₂C₂⋊₃₀C₂², C₄²⋊C₂⋊₉₆C₂², C₂³.₃₃₄(C₄○D₄), (C₂×C₄²).₉₂₉C₂², (C₂³×C₄).₅₉₈C₂², C₂.₁₉(C₂×2₊¹⁺⁴), (C₂²×C₄).₁₁₉₆C₂³, (C₂²×D₄).₅₉₂C₂², C₂².D₄⋊₄₄C₂², (C₂×C₄×D₄)⋊₈₅C₂, (C₂×C₄⋊D₄)⋊₆₃C₂, (C₂²×C₄⋊C₄)⋊₄₅C₂, C₄.₁₃₃(C₂×C₄○D₄), (C₂×C₄⋊C₄)⋊₁₃₇C₂², (C₂×C₄².C₂)⋊₄₃C₂, C₂².₃₃(C₂×C₄○D₄), C₂.₃₂(C₂²×C₄○D₄), (C₂×C₄²⋊₂C₂)⋊₃₆C₂, (C₂×C₄²⋊C₂)⋊₆₁C₂, (C₂×C₄).₈₄₉(C₄○D₄), (C₂×C₂².D₄)⋊₅₇C₂, (C₂×C₂²⋊C₄).₅₄₀C₂², SmallGroup(128,2203)

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

Derived series

C₁ — C₂² — C₂×C₂².₄₇C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄⋊C₄ — C₂×C₂².₄₇C₂⁴

Lower central

C₁ — C₂² — C₂×C₂².₄₇C₂⁴

Upper central

C₁ — C₂³ — C₂×C₂².₄₇C₂⁴

Jennings

C₁ — C₂² — C₂×C₂².₄₇C₂⁴

Generators and relations for C₂×C₂².₄₇C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=cb=bc, g²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede^-1=bd=db, geg^-1=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 956 in 626 conjugacy classes, 404 normal (32 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂×C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄⋊D₄, C₂².D₄, C₄².C₂, C₄²⋊₂C₂, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×C₄⋊C₄, C₂×C₄²⋊C₂, C₂×C₄×D₄, C₂×C₄×D₄, C₂×C₄⋊D₄, C₂×C₄⋊D₄, C₂×C₂².D₄, C₂×C₄².C₂, C₂×C₄²⋊₂C₂, C₂².₄₇C₂⁴, C₂×C₂².₄₇C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂².₄₇C₂⁴, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂×C₂².₄₇C₂⁴

Smallest permutation representation of C₂×C₂².₄₇C₂⁴
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2Q	4A	···	4T	4U	···	4AF
order	1	2	···	2	2	2	2	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	2	2	2	2	4	···	4	2	···	2	4	···	4

50 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂×C₂².₄₇C₂⁴

C₂²×C₄⋊C₄

C₂×C₄²⋊C₂

C₂×C₄×D₄

C₂×C₄⋊D₄

C₂×C₂².D₄

C₂×C₄².C₂

C₂×C₄²⋊₂C₂

C₂².₄₇C₂⁴

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂×C₂².₄₇C₂⁴ ►in GL₆(𝔽₅)

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

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

2	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	0	2
0	0	0	0	2	0

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

4	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	0	4
0	0	0	0	1	0

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

C₂×C₂².₄₇C₂⁴ in GAP, Magma, Sage, TeX

C_2\times C_2^2._{47}C_2^4

% in TeX

G:=Group("C2xC2^2.47C2^4");

// GroupNames label

G:=SmallGroup(128,2203);

// by ID

G=gap.SmallGroup(128,2203);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,570,136]);

// Polycyclic

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

// generators/relations

G = C2×C22.47C24 order 128 = 27

Direct product of C2 and C22.47C24

G = C₂×C₂².₄₇C₂⁴ order 128 = 2⁷

Direct product of C₂ and C₂².₄₇C₂⁴