Copied to
clipboard

G = S3xC4.Dic3order 288 = 25·32

Direct product of S3 and C4.Dic3

direct product, metabelian, supersoluble, monomial

Aliases: S3xC4.Dic3, C3:C8:19D6, (S3xC12).3C4, C12.92(C4xS3), (C4xS3).41D6, C3:5(S3xM4(2)), (C2xC12).289D6, (C3xS3):2M4(2), C62.43(C2xC4), (C4xS3).1Dic3, D6.7(C2xDic3), (C6xDic3).9C4, C4.14(S3xDic3), C32:5(C2xM4(2)), C12.58D6:7C2, D6.Dic3:13C2, (C6xC12).65C22, C12.16(C2xDic3), C6.2(C22xDic3), C22.6(S3xDic3), (S3xC12).57C22, (C3xC12).142C23, C12.141(C22xS3), Dic3.5(C2xDic3), C32:4C8:15C22, (C2xDic3).5Dic3, (C22xS3).3Dic3, (S3xC3:C8):12C2, C4.88(C2xS32), (C2xC4).61S32, (S3xC2xC6).6C4, (S3xC2xC4).1S3, C6.82(S3xC2xC4), (S3xC2xC12).4C2, C2.4(C2xS3xDic3), (C3xC3:C8):24C22, (C2xC6).72(C4xS3), C3:2(C2xC4.Dic3), (S3xC6).16(C2xC4), (C3xC12).55(C2xC4), (C3xC4.Dic3):9C2, (C2xC6).7(C2xDic3), (C3xC6).38(C22xC4), (C3xDic3).21(C2xC4), SmallGroup(288,461)

Series: Derived Chief Lower central Upper central

C1C3xC6 — S3xC4.Dic3
C1C3C32C3xC6C3xC12S3xC12S3xC3:C8 — S3xC4.Dic3
C32C3xC6 — S3xC4.Dic3
C1C4C2xC4

Generators and relations for S3xC4.Dic3
 G = < a,b,c,d,e | a3=b2=c4=1, d6=c2, e2=c2d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 370 in 146 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC8, M4(2), C22xC4, C3xS3, C3xS3, C3xC6, C3xC6, C3:C8, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C2xC12, C22xS3, C22xC6, C2xM4(2), C3xDic3, C3xC12, S3xC6, S3xC6, C62, S3xC8, C8:S3, C2xC3:C8, C4.Dic3, C4.Dic3, C3xM4(2), S3xC2xC4, C22xC12, C3xC3:C8, C32:4C8, S3xC12, C6xDic3, C6xC12, S3xC2xC6, S3xM4(2), C2xC4.Dic3, S3xC3:C8, D6.Dic3, C3xC4.Dic3, C12.58D6, S3xC2xC12, S3xC4.Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, Dic3, D6, M4(2), C22xC4, C4xS3, C2xDic3, C22xS3, C2xM4(2), S32, C4.Dic3, S3xC2xC4, C22xDic3, S3xDic3, C2xS32, S3xM4(2), C2xC4.Dic3, C2xS3xDic3, S3xC4.Dic3

Smallest permutation representation of S3xC4.Dic3
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(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)
(1 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 45 34 42 31 39 28 48)(26 38 35 47 32 44 29 41)(27 43 36 40 33 37 30 46)

G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (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), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,45,34,42,31,39,28,48)(26,38,35,47,32,44,29,41)(27,43,36,40,33,37,30,46)>;

G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (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), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,45,34,42,31,39,28,48)(26,38,35,47,32,44,29,41)(27,43,36,40,33,37,30,46) );

G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,36),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(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)], [(1,19,10,16,7,13,4,22),(2,24,11,21,8,18,5,15),(3,17,12,14,9,23,6,20),(25,45,34,42,31,39,28,48),(26,38,35,47,32,44,29,41),(27,43,36,40,33,37,30,46)]])

54 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A6B6C6D6E6F6G6H6I6J6K6L8A8B8C8D8E8F8G8H12A···12F12G···12K12L12M12N12O24A24B24C24D
order1222223334444446666666666668888888812···1212···121212121224242424
size1123362241123362222444466666666181818182···24···4666612121212

54 irreducible representations

dim111111111222222222222444444
type+++++++++-+-+-+-+-
imageC1C2C2C2C2C2C4C4C4S3S3D6Dic3D6Dic3D6Dic3M4(2)C4xS3C4xS3C4.Dic3S32S3xDic3C2xS32S3xDic3S3xM4(2)S3xC4.Dic3
kernelS3xC4.Dic3S3xC3:C8D6.Dic3C3xC4.Dic3C12.58D6S3xC2xC12S3xC12C6xDic3S3xC2xC6C4.Dic3S3xC2xC4C3:C8C4xS3C4xS3C2xDic3C2xC12C22xS3C3xS3C12C2xC6S3C2xC4C4C4C22C3C1
# reps122111422112221214228111124

Matrix representation of S3xC4.Dic3 in GL6(F73)

010000
72720000
001000
000100
000010
000001
,
7200000
110000
0072000
0007200
000010
000001
,
7200000
0720000
0027000
00724600
000010
000001
,
7200000
0720000
0027000
0002700
0000172
000010
,
2700000
0270000
00277100
00594600
0000027
0000270

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,72,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,59,0,0,0,0,71,46,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

S3xC4.Dic3 in GAP, Magma, Sage, TeX

S_3\times C_4.{\rm Dic}_3
% in TeX

G:=Group("S3xC4.Dic3");
// GroupNames label

G:=SmallGroup(288,461);
// by ID

G=gap.SmallGroup(288,461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,219,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^6=c^2,e^2=c^2*d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<