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G = C3xD6.Dic3order 432 = 24·33

Direct product of C3 and D6.Dic3

direct product, metabelian, supersoluble, monomial

Aliases: C3xD6.Dic3, C33:5M4(2), C12.103S32, D6.(C3xDic3), (S3xC6).4C12, (S3xC12).1C6, C6.29(S3xC12), C12.46(S3xC6), C6.2(C6xDic3), (S3xC12).13S3, C32:4C8:14C6, (C3xC12).180D6, Dic3.(C3xDic3), (S3xC6).5Dic3, C6.33(S3xDic3), (C3xDic3).1C12, C32:4(C3xM4(2)), C32:13(C8:S3), (C3xDic3).7Dic3, C32:8(C4.Dic3), (C32xDic3).4C4, (C32xC12).62C22, (C3xC3:C8):7C6, C3:C8:4(C3xS3), (C3xC3:C8):11S3, C4.15(C3xS32), (S3xC3xC6).4C4, C3:3(C3xC8:S3), (S3xC3xC12).2C2, C2.3(C3xS3xDic3), (C4xS3).2(C3xS3), (C32xC3:C8):14C2, (C3xC6).86(C4xS3), C3:1(C3xC4.Dic3), (C3xC12).63(C2xC6), (C3xC6).19(C2xC12), (C3xC32:4C8):18C2, (C32xC6).24(C2xC4), (C3xC6).46(C2xDic3), SmallGroup(432,416)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xD6.Dic3
C1C3C32C3xC6C3xC12C32xC12S3xC3xC12 — C3xD6.Dic3
C32C3xC6 — C3xD6.Dic3
C1C12

Generators and relations for C3xD6.Dic3
 G = < a,b,c,d,e | a3=b6=c2=1, d6=b3, e2=b3d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d5 >

Subgroups: 304 in 118 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, C32, C32, Dic3, C12, C12, D6, C2xC6, M4(2), C3xS3, C3xC6, C3xC6, C3:C8, C3:C8, C24, C4xS3, C2xC12, C33, C3xDic3, C3xDic3, C3xC12, C3xC12, S3xC6, S3xC6, C62, C8:S3, C4.Dic3, C3xM4(2), S3xC32, C32xC6, C3xC3:C8, C3xC3:C8, C32:4C8, C3xC24, S3xC12, S3xC12, C6xC12, C32xDic3, C32xC12, S3xC3xC6, D6.Dic3, C3xC8:S3, C3xC4.Dic3, C32xC3:C8, C3xC32:4C8, S3xC3xC12, C3xD6.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, Dic3, C12, D6, C2xC6, M4(2), C3xS3, C4xS3, C2xDic3, C2xC12, C3xDic3, S32, S3xC6, C8:S3, C4.Dic3, C3xM4(2), S3xDic3, S3xC12, C6xDic3, C3xS32, D6.Dic3, C3xC8:S3, C3xC4.Dic3, C3xS3xDic3, C3xD6.Dic3

Smallest permutation representation of C3xD6.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 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 16 18 20 22 24)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 48 46 44 42 40)
(1 35)(2 36)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(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 22 10 19 7 16 4 13)(2 15 11 24 8 21 5 18)(3 20 12 17 9 14 6 23)(25 46 34 43 31 40 28 37)(26 39 35 48 32 45 29 42)(27 44 36 41 33 38 30 47)

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

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

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

90 conjugacy classes

class 1 2A2B3A3B3C···3H3I3J3K4A4B4C6A6B6C···6H6I6J6K6L···6S8A8B8C8D12A12B12C12D12E···12P12Q···12V12W···12AD24A···24P24Q24R24S24T
order122333···3333444666···66666···688881212121212···1212···1212···1224···2424242424
size116112···2444116112···24446···666181811112···24···46···66···618181818

90 irreducible representations

dim111111111111222222222222222222444444
type++++++-+-+-
imageC1C2C2C2C3C4C4C6C6C6C12C12S3S3Dic3D6Dic3M4(2)C3xS3C3xS3C4xS3C3xDic3S3xC6C3xDic3C8:S3C4.Dic3C3xM4(2)S3xC12C3xC8:S3C3xC4.Dic3S32S3xDic3C3xS32D6.Dic3C3xS3xDic3C3xD6.Dic3
kernelC3xD6.Dic3C32xC3:C8C3xC32:4C8S3xC3xC12D6.Dic3C32xDic3S3xC3xC6C3xC3:C8C32:4C8S3xC12C3xDic3S3xC6C3xC3:C8S3xC12C3xDic3C3xC12S3xC6C33C3:C8C4xS3C3xC6Dic3C12D6C32C32C32C6C3C3C12C6C4C3C2C1
# reps111122222244111212222242444488112224

Matrix representation of C3xD6.Dic3 in GL6(F73)

6400000
0640000
008000
000800
000010
000001
,
7200000
0720000
0007200
0017200
000010
000001
,
57600000
14160000
0072000
0072100
000010
000001
,
2700000
0270000
001000
000100
0000721
0000720
,
0270000
100000
0072000
0007200
000001
000010

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,14,0,0,0,0,60,16,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3xD6.Dic3 in GAP, Magma, Sage, TeX

C_3\times D_6.{\rm Dic}_3
% in TeX

G:=Group("C3xD6.Dic3");
// GroupNames label

G:=SmallGroup(432,416);
// by ID

G=gap.SmallGroup(432,416);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,2028,14118]);
// Polycyclic

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

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