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G = C3xD6:D6order 432 = 24·33

Direct product of C3 and D6:D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xD6:D6, C12:8S32, C12:1(S3xC6), D6:2(S3xC6), (S3xC6):12D6, (C3xD12):7C6, (C3xD12):5S3, D12:4(C3xS3), (C3xC12):10D6, C32:6(C6xD4), C33:12(C2xD4), D6:S3:3C6, C32:22(S3xD4), (C32xD12):9C2, (C32xC12):2C22, (C32xC6).28C23, C4:2(C3xS32), C3:2(C3xS3xD4), (S32xC6):6C2, (C2xS32):2C6, C6.9(S3xC2xC6), (C4xC3:S3):7C6, C2.11(S32xC6), (C3xC3:S3):7D4, C3:S3:3(C3xD4), (C12xC3:S3):4C2, (S3xC6):2(C2xC6), C6.112(C2xS32), (C3xC12):6(C2xC6), (S3xC3xC6):8C22, C3:Dic3:7(C2xC6), (C3xD6:S3):10C2, (C6xC3:S3).48C22, (C3xC6).19(C22xC6), (C3xC6).133(C22xS3), (C3xC3:Dic3):22C22, (C2xC3:S3).20(C2xC6), SmallGroup(432,650)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xD6:D6
C1C3C32C3xC6C32xC6S3xC3xC6S32xC6 — C3xD6:D6
C32C3xC6 — C3xD6:D6
C1C6C12

Generators and relations for C3xD6:D6
 G = < a,b,c,d,e | a3=b6=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=b3c, ede=d-1 >

Subgroups: 1056 in 270 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2xC4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2xC6, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C33, C3xDic3, C3:Dic3, C3xC12, C3xC12, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C62, S3xD4, C6xD4, S3xC32, C3xC3:S3, C32xC6, D6:S3, S3xC12, C3xD12, C3xD12, C3xC3:D4, C4xC3:S3, D4xC32, C2xS32, S3xC2xC6, C3xC3:Dic3, C32xC12, C3xS32, S3xC3xC6, C6xC3:S3, D6:D6, C3xS3xD4, C3xD6:S3, C32xD12, C12xC3:S3, S32xC6, C3xD6:D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3xD4, C22xS3, C22xC6, S32, S3xC6, S3xD4, C6xD4, C2xS32, S3xC2xC6, C3xS32, D6:D6, C3xS3xD4, S32xC6, C3xD6:D6

Smallest permutation representation of C3xD6:D6
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 39 41)(38 40 42)(43 45 47)(44 46 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)(2 15)(3 14)(4 13)(5 18)(6 17)(7 22)(8 21)(9 20)(10 19)(11 24)(12 23)(25 40)(26 39)(27 38)(28 37)(29 42)(30 41)(31 46)(32 45)(33 44)(34 43)(35 48)(36 47)
(1 21 5 23 3 19)(2 20 6 22 4 24)(7 18 11 14 9 16)(8 17 12 13 10 15)(25 46 27 44 29 48)(26 45 28 43 30 47)(31 37 33 41 35 39)(32 42 34 40 36 38)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 30)(8 25)(9 26)(10 27)(11 28)(12 29)(13 44)(14 45)(15 46)(16 47)(17 48)(18 43)(19 41)(20 42)(21 37)(22 38)(23 39)(24 40)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H3I3J3K4A4B6A6B6C···6H6I6J6K6L···6S6T6U6V6W6X···6AI12A12B12C···12N12O12P
order12222222333···333344666···66666···666666···6121212···121212
size11666699112···2444218112···24446···6999912···12224···41818

72 irreducible representations

dim11111111112222222244444444
type++++++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D6D6C3xS3C3xD4S3xC6S3xC6S32S3xD4C2xS32C3xS32D6:D6C3xS3xD4S32xC6C3xD6:D6
kernelC3xD6:D6C3xD6:S3C32xD12C12xC3:S3S32xC6D6:D6D6:S3C3xD12C4xC3:S3C2xS32C3xD12C3xC3:S3C3xC12S3xC6D12C3:S3C12D6C12C32C6C4C3C3C2C1
# reps12212244242224444812122424

Matrix representation of C3xD6:D6 in GL6(F13)

300000
030000
009000
000900
000010
000001
,
1200000
0120000
001000
000100
00001212
000010
,
720000
260000
0012000
0001200
00001212
000001
,
1170000
720000
00121200
001000
000010
00001212
,
1170000
720000
0012000
001100
000010
000001

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[7,2,0,0,0,0,2,6,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1],[11,7,0,0,0,0,7,2,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[11,7,0,0,0,0,7,2,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3xD6:D6 in GAP, Magma, Sage, TeX

C_3\times D_6\rtimes D_6
% in TeX

G:=Group("C3xD6:D6");
// GroupNames label

G:=SmallGroup(432,650);
// by ID

G=gap.SmallGroup(432,650);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,303,142,2028,14118]);
// Polycyclic

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

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