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G = S3×C4.Dic3order 288 = 25·32

Direct product of S3 and C4.Dic3

direct product, metabelian, supersoluble, monomial

Aliases: S3×C4.Dic3, C3⋊C819D6, (S3×C12).3C4, C12.92(C4×S3), (C4×S3).41D6, C35(S3×M4(2)), (C2×C12).289D6, (C3×S3)⋊2M4(2), C62.43(C2×C4), (C4×S3).1Dic3, D6.7(C2×Dic3), (C6×Dic3).9C4, C4.14(S3×Dic3), C325(C2×M4(2)), C12.58D67C2, D6.Dic313C2, (C6×C12).65C22, C12.16(C2×Dic3), C6.2(C22×Dic3), C22.6(S3×Dic3), (S3×C12).57C22, (C3×C12).142C23, C12.141(C22×S3), Dic3.5(C2×Dic3), C324C815C22, (C2×Dic3).5Dic3, (C22×S3).3Dic3, (S3×C3⋊C8)⋊12C2, C4.88(C2×S32), (C2×C4).61S32, (S3×C2×C6).6C4, (S3×C2×C4).1S3, C6.82(S3×C2×C4), (S3×C2×C12).4C2, C2.4(C2×S3×Dic3), (C3×C3⋊C8)⋊24C22, (C2×C6).72(C4×S3), C32(C2×C4.Dic3), (S3×C6).16(C2×C4), (C3×C12).55(C2×C4), (C3×C4.Dic3)⋊9C2, (C2×C6).7(C2×Dic3), (C3×C6).38(C22×C4), (C3×Dic3).21(C2×C4), SmallGroup(288,461)

Series: Derived Chief Lower central Upper central

C1C3×C6 — S3×C4.Dic3
C1C3C32C3×C6C3×C12S3×C12S3×C3⋊C8 — S3×C4.Dic3
C32C3×C6 — S3×C4.Dic3
C1C4C2×C4

Generators and relations for S3×C4.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 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], S3, C6 [×2], C6 [×7], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8 [×2], M4(2) [×4], C22×C4, C3×S3 [×2], C3×S3, C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8 [×2], C4.Dic3, C4.Dic3 [×5], C3×M4(2), S3×C2×C4, C22×C12, C3×C3⋊C8 [×2], C324C8 [×2], S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, S3×M4(2), C2×C4.Dic3, S3×C3⋊C8 [×2], D6.Dic3 [×2], C3×C4.Dic3, C12.58D6, S3×C2×C12, S3×C4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], M4(2) [×2], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C2×M4(2), S32, C4.Dic3 [×2], S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, S3×M4(2), C2×C4.Dic3, C2×S3×Dic3, S3×C4.Dic3

Smallest permutation representation of S3×C4.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 16 10 13 7 22 4 19)(2 21 11 18 8 15 5 24)(3 14 12 23 9 20 6 17)(25 42 34 39 31 48 28 45)(26 47 35 44 32 41 29 38)(27 40 36 37 33 46 30 43)

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim111111111222222222222444444
type+++++++++-+-+-+-+-
imageC1C2C2C2C2C2C4C4C4S3S3D6Dic3D6Dic3D6Dic3M4(2)C4×S3C4×S3C4.Dic3S32S3×Dic3C2×S32S3×Dic3S3×M4(2)S3×C4.Dic3
kernelS3×C4.Dic3S3×C3⋊C8D6.Dic3C3×C4.Dic3C12.58D6S3×C2×C12S3×C12C6×Dic3S3×C2×C6C4.Dic3S3×C2×C4C3⋊C8C4×S3C4×S3C2×Dic3C2×C12C22×S3C3×S3C12C2×C6S3C2×C4C4C4C22C3C1
# reps122111422112221214228111124

Matrix representation of S3×C4.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] >;

S3×C4.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

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