Copied to
clipboard

G = C24.584C23order 128 = 27

65th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.584C23, C23.469C24, C22.2522+ 1+4, C23.99(C2×Q8), (C22×C4).59Q8, (C2×C42).66C22, C2.5(C232Q8), C23.315(C4○D4), (C23×C4).406C22, (C22×C4).544C23, C23.8Q8.31C2, C23.7Q8.53C2, C22.110(C22×Q8), C22.15(C42.C2), C23.83C2341C2, C23.63C2389C2, C23.81C2342C2, C2.52(C22.45C24), C2.C42.205C22, C2.28(C23.37C23), C2.59(C22.47C24), (C2×C4).165(C2×Q8), (C4×C22⋊C4).63C2, C2.16(C2×C42.C2), (C2×C4).393(C4○D4), (C2×C4⋊C4).316C22, C22.345(C2×C4○D4), (C2×C22⋊C4).469C22, (C2×C2.C42).29C2, SmallGroup(128,1301)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.584C23
C1C2C22C23C24C23×C4C2×C2.C42 — C24.584C23
C1C23 — C24.584C23
C1C23 — C24.584C23
C1C23 — C24.584C23

Generators and relations for C24.584C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=cb=bc, e2=ca=ac, f2=b, ab=ba, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 420 in 228 conjugacy classes, 104 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×18], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×8], C2×C4 [×50], C23, C23 [×6], C23 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C24, C2.C42 [×4], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×10], C23×C4, C23×C4 [×2], C2×C2.C42, C4×C22⋊C4, C23.7Q8, C23.7Q8 [×2], C23.8Q8 [×2], C23.63C23 [×4], C23.81C23 [×2], C23.83C23 [×2], C24.584C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×4], 2+ 1+4 [×2], C2×C42.C2, C23.37C23, C232Q8, C22.45C24 [×2], C22.47C24 [×2], C24.584C23

Smallest permutation representation of C24.584C23
On 64 points
Generators in S64
(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 47 39 59)(2 16 40 32)(3 45 37 57)(4 14 38 30)(5 29 21 13)(6 58 22 46)(7 31 23 15)(8 60 24 48)(9 33 25 17)(10 62 26 50)(11 35 27 19)(12 64 28 52)(18 42 34 54)(20 44 36 56)(41 61 53 49)(43 63 55 51)
(1 9 5 55)(2 56 6 10)(3 11 7 53)(4 54 8 12)(13 19 59 61)(14 62 60 20)(15 17 57 63)(16 64 58 18)(21 43 39 25)(22 26 40 44)(23 41 37 27)(24 28 38 42)(29 35 47 49)(30 50 48 36)(31 33 45 51)(32 52 46 34)
(1 7)(2 40)(3 5)(4 38)(6 22)(8 24)(9 53)(10 26)(11 55)(12 28)(13 57)(14 30)(15 59)(16 32)(17 61)(18 34)(19 63)(20 36)(21 37)(23 39)(25 41)(27 43)(29 45)(31 47)(33 49)(35 51)(42 54)(44 56)(46 58)(48 60)(50 62)(52 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4V4W4X4Y4Z
order12···2222244444···44444
size11···1222222224···48888

38 irreducible representations

dim111111112224
type++++++++-+
imageC1C2C2C2C2C2C2C2Q8C4○D4C4○D42+ 1+4
kernelC24.584C23C2×C2.C42C4×C22⋊C4C23.7Q8C23.8Q8C23.63C23C23.81C23C23.83C23C22×C4C2×C4C23C22
# reps111324224882

Matrix representation of C24.584C23 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
100000
000100
004000
000001
000010
,
200000
030000
002000
000300
000002
000020
,
400000
040000
002000
000300
000001
000040
,
100000
040000
001000
000400
000040
000004

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

C24.584C23 in GAP, Magma, Sage, TeX

C_2^4._{584}C_2^3
% in TeX

G:=Group("C2^4.584C2^3");
// GroupNames label

G:=SmallGroup(128,1301);
// by ID

G=gap.SmallGroup(128,1301);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,184,675]);
// Polycyclic

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

׿
×
𝔽