C2^4.438C2^3 - GroupNames

G = C₂⁴.₄₃₈C₂³ order 128 = 2⁷

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

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

Aliases: C₂⁴.₄₃₈C₂³, C₂³.₆₅₇C₂⁴, C₂².₄₃₀2₊¹⁺⁴, C₂².₃₂₄2_-¹⁺⁴, C₄²⋊₅C₄⋊₃₁C₂, C₂³.₉₄(C₄○D₄), C₂³.Q₈⋊₈₁C₂, (C₂×C₄²).₆₉₃C₂², (C₂²×C₄).₅₇₇C₂³, (C₂³×C₄).₁₆₅C₂², C₂³.₈Q₈⋊₁₃₀C₂, C₂³.₁₁D₄⋊₁₁₄C₂, C₂³.₂₃D₄.₆₈C₂, C₂³.₁₀D₄.₅₉C₂, (C₂²×D₄).₂₇₂C₂², C₂⁴.C₂²⋊₁₆₃C₂, C₂⁴.₃C₂².₇₁C₂, C₂³.₆₃C₂³⋊₁₆₉C₂, C₂³.₆₅C₂³⋊₁₄₂C₂, C₂³.₈₁C₂³⋊₁₁₄C₂, C₂.₂₉(C₂².₅₄C₂⁴), C₂.C₄².₃₆₁C₂², C₂.₁₀₉(C₂².₄₅C₂⁴), C₂.₉₉(C₂².₄₇C₂⁴), C₂.₉₉(C₂².₃₆C₂⁴), C₂.₉₁(C₂².₃₃C₂⁴), (C₂×C₄).₂₁₈(C₄○D₄), (C₂×C₄⋊C₄).₄₆₈C₂², C₂².₅₁₈(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₃₀₇C₂², SmallGroup(128,1489)

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

Derived series

C₁ — C₂³ — C₂⁴.₄₃₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂⁴.₃C₂² — C₂⁴.₄₃₈C₂³

Lower central

C₁ — C₂³ — C₂⁴.₄₃₈C₂³

Upper central

C₁ — C₂³ — C₂⁴.₄₃₈C₂³

Jennings

C₁ — C₂³ — C₂⁴.₄₃₈C₂³

Generators and relations for C₂⁴.₄₃₈C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=1, e²=db=bd, f²=cb=bc, g²=b, eae^-1=gag^-1=ab=ba, ac=ca, faf^-1=ad=da, 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: 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, C₄²⋊₅C₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂³.₆₃C₂³, C₂⁴.C₂², C₂³.₆₅C₂³, C₂⁴.₃C₂², C₂³.₁₀D₄, C₂³.Q₈, C₂³.₁₁D₄, C₂³.₈₁C₂³, C₂⁴.₄₃₈C₂³
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₆C₂⁴, C₂².₄₅C₂⁴, C₂².₄₇C₂⁴, C₂².₅₄C₂⁴, C₂⁴.₄₃₈C₂³

Smallest permutation representation of C₂⁴.₄₃₈C₂³
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

C₂⁴.₄₃₈C₂³

C₄²⋊₅C₄

C₂³.₈Q₈

C₂³.₂₃D₄

C₂³.₆₃C₂³

C₂⁴.C₂²

C₂³.₆₅C₂³

C₂⁴.₃C₂²

C₂³.₁₀D₄

C₂³.Q₈

C₂³.₁₁D₄

C₂³.₈₁C₂³

C₂×C₄

C₂³

C₂²

# reps

Matrix representation of C₂⁴.₄₃₈C₂³ ►in GL₆(𝔽₅)

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

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

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

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

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

C₂⁴.₄₃₈C₂³ in GAP, Magma, Sage, TeX

C_2^4._{438}C_2^3

% in TeX

G:=Group("C2^4.438C2^3");

// GroupNames label

G:=SmallGroup(128,1489);

// by ID

G=gap.SmallGroup(128,1489);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,268,1571,346,80]);

// 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*b=b*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*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 = C24.438C23 order 128 = 27

278th non-split extension by C24 of C23 acting via C23/C2=C22

G = C₂⁴.₄₃₈C₂³ order 128 = 2⁷

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