C2^4.440C2^3 - GroupNames

G = C₂⁴.₄₄₀C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₄₀C₂³, C₂³.₆₆₁C₂⁴, C₂².₃₂₇2_-¹⁺⁴, C₂².₄₃₄2₊¹⁺⁴, C₄²⋊₈C₄⋊₆₁C₂, C₂³.Q₈⋊₈₂C₂, C₂³.₁₉₁(C₄○D₄), C₂³.₃₄D₄⋊₆₀C₂, (C₂³×C₄).₁₆₇C₂², (C₂×C₄²).₁₀₂C₂², (C₂²×C₄).₅₈₁C₂³, C₂³.₈Q₈⋊₁₃₁C₂, C₂³.₁₁D₄⋊₁₁₅C₂, C₂³.₁₀D₄.₆₀C₂, C₂³.₂₃D₄.₆₉C₂, (C₂²×D₄).₂₇₄C₂², C₂⁴.C₂²⋊₁₆₄C₂, C₂.₉₀(C₂².₃₂C₂⁴), C₂⁴.₃C₂².₇₂C₂, C₂³.₈₁C₂³⋊₁₁₅C₂, C₂³.₆₃C₂³⋊₁₇₁C₂, C₂.C₄².₃₆₅C₂², C₂.₁₁₃(C₂².₄₅C₂⁴), C₂.₃₅(C₂².₅₆C₂⁴), C₂.₅₇(C₂².₃₄C₂⁴), C₂.₁₀₀(C₂².₄₇C₂⁴), C₂.₁₀₀(C₂².₄₆C₂⁴), C₂.₁₀₂(C₂².₃₆C₂⁴), (C₂×C₄).₂₂₀(C₄○D₄), (C₂×C₄⋊C₄).₄₇₁C₂², C₂².₅₂₂(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₇₀C₂², SmallGroup(128,1493)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₄₀C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂³.₂₃D₄ — C₂⁴.₄₄₀C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₄₀C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₄₀C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₄₀C₂³

Generators and relations for C₂⁴.₄₄₀C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=dc=cd, f²=b, g²=cb=bc, faf^-1=ab=ba, ac=ca, ad=da, eae^-1=abc, ag=ga, bd=db, fef^-1=be=eb, bf=fb, bg=gb, geg^-1=ce=ec, cf=fc, cg=gc, de=ed, gfg^-1=df=fd, dg=gd >

Subgroups: 452 in 217 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, C₂³.₃₄D₄, C₄²⋊₈C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂⁴.₄₄₀C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₂C₂⁴, C₂².₃₄C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₆C₂⁴, C₂².₄₇C₂⁴, C₂².₅₆C₂⁴, C₂⁴.₄₄₀C₂³

Smallest permutation representation of C₂⁴.₄₄₀C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

C₄○D₄

2₊¹⁺⁴

2_-¹⁺⁴

kernel

C₂⁴.₄₄₀C₂³

C₂³.₃₄D₄

C₄²⋊₈C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₄₀C₂³ ►in GL₆(𝔽₅)

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

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

0	4	0	0	0	0
1	0	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

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

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

C₂⁴.₄₄₀C₂³ in GAP, Magma, Sage, TeX

C_2^4._{440}C_2^3

% in TeX

G:=Group("C2^4.440C2^3");

// GroupNames label

G:=SmallGroup(128,1493);

// by ID

G=gap.SmallGroup(128,1493);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,268,1571,346]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d*c=c*d,f^2=b,g^2=c*b=b*c,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,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 = C24.440C23 order 128 = 27

280th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₄₀C₂³ order 128 = 2⁷

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