C2^4.437C2^3 - GroupNames

G = C₂⁴.₄₃₇C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₃₇C₂³, C₂³.₆₅₃C₂⁴, C₂².₄₂₆2₊¹⁺⁴, C₄²⋊₈C₄⋊₅₉C₂, C₂³.₉₂(C₄○D₄), C₂³⋊₂D₄.₃₀C₂, (C₂×C₄²).₉₈C₂², (C₂³×C₄).₁₆₄C₂², (C₂²×C₄).₅₇₄C₂³, C₂³.₈Q₈⋊₁₂₉C₂, C₂³.₁₀D₄⋊₁₀₁C₂, C₂³.₂₃D₄⋊₁₀₃C₂, C₂³.₁₁D₄⋊₁₁₂C₂, C₂⁴.₃C₂²⋊₉₃C₂, C₂.₉(C₂⁴⋊C₂²), (C₂²×D₄).₂₇₀C₂², C₂⁴.C₂²⋊₁₆₁C₂, C₂.₈₆(C₂².₃₂C₂⁴), C₂³.₈₃C₂³⋊₁₀₃C₂, C₂.C₄².₃₅₇C₂², C₂.₁₀₅(C₂².₄₅C₂⁴), C₂.₉₇(C₂².₄₇C₂⁴), C₂.₅₅(C₂².₃₄C₂⁴), (C₂×C₄).₄₅₂(C₄○D₄), (C₂×C₄⋊C₄).₄₆₄C₂², C₂².₅₁₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₆₈C₂², SmallGroup(128,1485)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₃₇C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂⁴.₄₃₇C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₃₇C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₃₇C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₃₇C₂³

Generators and relations for C₂⁴.₄₃₇C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=db=bd, f²=c, g²=b, eae^-1=gag^-1=ab=ba, ac=ca, faf^-1=ad=da, bc=cb, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=bce, fg=gf >

Subgroups: 548 in 242 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, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂⁴.C₂², C₂⁴.₃C₂², C₂³⋊₂D₄, C₂³.₁₀D₄, C₂³.₁₁D₄, C₂³.₈₃C₂³, C₂⁴.₄₃₇C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂⁴⋊C₂², C₂⁴.₄₃₇C₂³

Smallest permutation representation of C₂⁴.₄₃₇C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4P	4Q	4R	4S	4T
order	1	2	···	2	2	2	2	2	4	···	4	4	4	4	4
size	1	1	···	1	4	4	8	8	4	···	4	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

kernel

C₂⁴.₄₃₇C₂³

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³⋊₂D₄

C₂³.₁₀D₄

C₂³.₁₁D₄

C₂³.₈₃C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₃₇C₂³ ►in GL₆(𝔽₅)

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

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	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

3	3	0	0	0	0
0	2	0	0	0	0
0	0	0	2	0	0
0	0	3	0	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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

3	0	0	0	0	0
4	2	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

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

C₂⁴.₄₃₇C₂³ in GAP, Magma, Sage, TeX

C_2^4._{437}C_2^3

% in TeX

G:=Group("C2^4.437C2^3");

// GroupNames label

G:=SmallGroup(128,1485);

// by ID

G=gap.SmallGroup(128,1485);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,184,1571,346,80]);

// Polycyclic

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

// generators/relations

G = C24.437C23 order 128 = 27

277th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₃₇C₂³ order 128 = 2⁷

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