Copied to
clipboard

G = C3xDic3.D6order 432 = 24·33

Direct product of C3 and Dic3.D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xDic3.D6, C12.100S32, C33:8(C2xQ8), C32:5(C6xQ8), C12.38(S3xC6), (C3xDic6):5S3, (C3xDic6):6C6, Dic6:4(C3xS3), C32:2Q8:3C6, C32:12(S3xQ8), (C3xC12).137D6, Dic3.2(S3xC6), C6.D6.1C6, (C3xDic3).25D6, (C32xDic6):8C2, (C32xC6).23C23, (C32xC12).38C22, (C32xDic3).10C22, C3:1(C3xS3xQ8), C2.7(S32xC6), C6.4(S3xC2xC6), C4.12(C3xS32), (C3xC3:S3):4Q8, C3:S3:3(C3xQ8), C6.107(C2xS32), (C4xC3:S3).4C6, (C12xC3:S3).5C2, (C3xC12).53(C2xC6), (C3xC32:2Q8):9C2, (C6xC3:S3).45C22, C3:Dic3.17(C2xC6), (C3xC6).14(C22xC6), (C3xDic3).3(C2xC6), (C3xC6.D6).2C2, (C3xC6).128(C22xS3), (C3xC3:Dic3).54C22, (C2xC3:S3).17(C2xC6), SmallGroup(432,645)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C3xDic3.D6
C1C3C32C3xC6C32xC6C32xDic3C3xC6.D6 — C3xDic3.D6
C32C3xC6 — C3xDic3.D6
C1C6C12

Generators and relations for C3xDic3.D6
 G = < a,b,c,d,e | a3=b6=1, c2=d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=b3d5 >

Subgroups: 608 in 198 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2xC4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2xC6, C2xQ8, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, Dic6, Dic6, C4xS3, C2xC12, C3xQ8, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, C3xC12, C3xC12, S3xC6, C2xC3:S3, S3xQ8, C6xQ8, C3xC3:S3, C32xC6, C6.D6, C32:2Q8, C3xDic6, C3xDic6, S3xC12, C4xC3:S3, Q8xC32, C32xDic3, C3xC3:Dic3, C32xC12, C6xC3:S3, Dic3.D6, C3xS3xQ8, C3xC6.D6, C3xC32:2Q8, C32xDic6, C12xC3:S3, C3xDic3.D6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2xC6, C2xQ8, C3xS3, C3xQ8, C22xS3, C22xC6, S32, S3xC6, S3xQ8, C6xQ8, C2xS32, S3xC2xC6, C3xS32, Dic3.D6, C3xS3xQ8, S32xC6, C3xDic3.D6

Smallest permutation representation of C3xDic3.D6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 19 7 13)(2 14 8 20)(3 21 9 15)(4 16 10 22)(5 23 11 17)(6 18 12 24)(25 38 31 44)(26 45 32 39)(27 40 33 46)(28 47 34 41)(29 42 35 48)(30 37 36 43)
(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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K4A4B4C4D4E4F6A6B6C···6H6I6J6K6L6M6N6O12A12B12C···12N12O···12V12W···12AH12AI12AJ
order1222333···3333444444666···66666666121212···1212···1212···121212
size1199112···24442666618112···24449999224···46···612···121818

72 irreducible representations

dim11111111112222222244444444
type++++++-+++-+
imageC1C2C2C2C2C3C6C6C6C6S3Q8D6D6C3xS3C3xQ8S3xC6S3xC6S32S3xQ8C2xS32C3xS32Dic3.D6C3xS3xQ8S32xC6C3xDic3.D6
kernelC3xDic3.D6C3xC6.D6C3xC32:2Q8C32xDic6C12xC3:S3Dic3.D6C6.D6C32:2Q8C3xDic6C4xC3:S3C3xDic6C3xC3:S3C3xDic3C3xC12Dic6C3:S3Dic3C12C12C32C6C4C3C3C2C1
# reps12221244422242448412122424

Matrix representation of C3xDic3.D6 in GL6(F13)

300000
030000
009000
000900
000010
000001
,
1200000
0120000
001000
000100
0000012
0000112
,
010000
1200000
0012000
0001200
000001
000010
,
800000
050000
0012100
0012000
000010
000001
,
080000
800000
0001200
0012000
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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3xDic3.D6 in GAP, Magma, Sage, TeX

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

G:=Group("C3xDic3.D6");
// GroupNames label

G:=SmallGroup(432,645);
// by ID

G=gap.SmallGroup(432,645);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,176,590,303,142,2028,14118]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<