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G = C3xS3xD12order 432 = 24·33

Direct product of C3, S3 and D12

direct product, metabelian, supersoluble, monomial

Aliases: C3xS3xD12, C12:6S32, C12:3(S3xC6), D6:1(S3xC6), C3:1(C6xD12), (S3xC12):4C6, (S3xC12):4S3, (S3xC6):11D6, (C3xD12):6C6, (C3xC12):19D6, C32:5(C6xD4), C33:11(C2xD4), C3:D12:1C6, Dic3:3(S3xC6), (S3xC32):4D4, C12:S3:11C6, C32:20(S3xD4), (C3xDic3):17D6, (C32xD12):8C2, C32:12(C2xD12), (C32xC12):1C22, (C32xC6).27C23, (C32xDic3):14C22, C4:1(C3xS32), C3:1(C3xS3xD4), (C2xS32):1C6, (S32xC6):5C2, C6.8(S3xC2xC6), (S3xC3xC12):5C2, C2.10(S32xC6), (C4xS3):3(C3xS3), (S3xC6):1(C2xC6), C6.111(C2xS32), (C3xC12):5(C2xC6), (C3xS3):1(C3xD4), (S3xC3xC6):7C22, (C6xC3:S3):6C22, (C3xC12:S3):7C2, (C3xDic3):3(C2xC6), (C3xC3:D12):13C2, (C3xC6).18(C22xC6), (C3xC6).132(C22xS3), (C2xC3:S3):4(C2xC6), SmallGroup(432,649)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xS3xD12
C1C3C32C3xC6C32xC6S3xC3xC6S32xC6 — C3xS3xD12
C32C3xC6 — C3xS3xD12
C1C6C12

Generators and relations for C3xS3xD12
 G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1096 in 270 conjugacy classes, 72 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2xC4, D4, C23, C32, C32, Dic3, C12, C12, D6, D6, D6, C2xC6, C2xD4, C3xS3, C3xS3, C3:S3, C3xC6, C3xC6, C4xS3, D12, D12, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C33, C3xDic3, C3xDic3, C3xC12, C3xC12, S32, S3xC6, S3xC6, S3xC6, C2xC3:S3, C62, C2xD12, S3xD4, C6xD4, S3xC32, S3xC32, C3xC3:S3, C32xC6, C3:D12, S3xC12, S3xC12, C3xD12, C3xD12, C3xC3:D4, C12:S3, C6xC12, D4xC32, C2xS32, S3xC2xC6, C32xDic3, C32xC12, C3xS32, S3xC3xC6, S3xC3xC6, C6xC3:S3, S3xD12, C6xD12, C3xS3xD4, C3xC3:D12, S3xC3xC12, C32xD12, C3xC12:S3, S32xC6, C3xS3xD12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, D12, C3xD4, C22xS3, C22xC6, S32, S3xC6, C2xD12, S3xD4, C6xD4, C3xD12, C2xS32, S3xC2xC6, C3xS32, S3xD12, C6xD12, C3xS3xD4, S32xC6, C3xS3xD12

Smallest permutation representation of C3xS3xD12
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 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 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 48)(36 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), (1,9,5)(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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)>;

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

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

81 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H3I3J3K4A4B6A6B6C···6H6I6J6K6L6M6N6O6P···6Y6Z···6AE6AF6AG6AH6AI12A···12H12I···12Q12R···12Y
order12222222333···333344666···666666666···66···6666612···1212···1212···12
size1133661818112···244426112···233334446···612···12181818182···24···46···6

81 irreducible representations

dim1111111111112222222222222244444444
type+++++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3S3D4D6D6D6C3xS3C3xS3D12C3xD4S3xC6S3xC6S3xC6C3xD12S32S3xD4C2xS32C3xS32S3xD12C3xS3xD4S32xC6C3xS3xD12
kernelC3xS3xD12C3xC3:D12S3xC3xC12C32xD12C3xC12:S3S32xC6S3xD12C3:D12S3xC12C3xD12C12:S3C2xS32S3xC12C3xD12S3xC32C3xDic3C3xC12S3xC6C4xS3D12C3xS3C3xS3Dic3C12D6S3C12C32C6C4C3C3C2C1
# reps1211122422241121232244246811122224

Matrix representation of C3xS3xD12 in GL6(F13)

900000
090000
003000
000300
000010
000001
,
100000
010000
000100
00121200
000010
000001
,
1200000
0120000
0001200
0012000
000010
000001
,
800000
050000
0012000
0001200
0000121
0000120
,
050000
800000
001000
000100
0000120
0000121

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

C3xS3xD12 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_{12}
% in TeX

G:=Group("C3xS3xD12");
// GroupNames label

G:=SmallGroup(432,649);
// by ID

G=gap.SmallGroup(432,649);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,142,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,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,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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