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G = C3xDic32order 432 = 24·33

Direct product of C3, Dic3 and Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3xDic32, C33:4C42, C62.102D6, C6.30(S3xC12), C3:Dic3:3C12, C32:3(C4xC12), C3:1(Dic3xC12), C6.3(C6xDic3), (C3xDic3):2C12, C62.17(C2xC6), (C6xDic3).9C6, C6.34(S3xDic3), C32:8(C4xDic3), (C6xDic3).20S3, (C32xDic3):5C4, (C3xC62).1C22, C6.25(C6.D6), (C2xC6).66S32, C22.3(C3xS32), C2.2(C3xS3xDic3), (C3xC6).60(C4xS3), (C2xC6).21(S3xC6), (C3xC3:Dic3):1C4, (C3xC6).21(C2xC12), (C2xC3:Dic3).5C6, (C6xC3:Dic3).1C2, (Dic3xC3xC6).10C2, C2.2(C3xC6.D6), (C32xC6).26(C2xC4), (C2xDic3).4(C3xS3), (C3xC6).47(C2xDic3), SmallGroup(432,425)

Series: Derived Chief Lower central Upper central

C1C32 — C3xDic32
C1C3C32C3xC6C62C3xC62Dic3xC3xC6 — C3xDic32
C32 — C3xDic32
C1C2xC6

Generators and relations for C3xDic32
 G = < a,b,c,d,e | a3=b6=d6=1, c2=b3, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 448 in 178 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2xC4, C32, C32, C32, Dic3, Dic3, C12, C2xC6, C2xC6, C2xC6, C42, C3xC6, C3xC6, C3xC6, C2xDic3, C2xDic3, C2xC12, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, C62, C62, C62, C4xDic3, C4xC12, C32xC6, C32xC6, C6xDic3, C6xDic3, C2xC3:Dic3, C6xC12, C32xDic3, C3xC3:Dic3, C3xC62, Dic32, Dic3xC12, Dic3xC3xC6, C6xC3:Dic3, C3xDic32
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, Dic3, C12, D6, C2xC6, C42, C3xS3, C4xS3, C2xDic3, C2xC12, C3xDic3, S32, S3xC6, C4xDic3, C4xC12, S3xDic3, C6.D6, S3xC12, C6xDic3, C3xS32, Dic32, Dic3xC12, C3xS3xDic3, C3xC6.D6, C3xDic32

Smallest permutation representation of C3xDic32
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(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 30 4 27)(2 29 5 26)(3 28 6 25)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 20 16 23)(14 19 17 22)(15 24 18 21)(37 44 40 47)(38 43 41 46)(39 48 42 45)
(1 14 5 18 3 16)(2 15 6 13 4 17)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 26 21 28 23 30)(20 27 22 29 24 25)(31 41 33 37 35 39)(32 42 34 38 36 40)
(1 42 18 36)(2 37 13 31)(3 38 14 32)(4 39 15 33)(5 40 16 34)(6 41 17 35)(7 27 48 24)(8 28 43 19)(9 29 44 20)(10 30 45 21)(11 25 46 22)(12 26 47 23)

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

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

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

108 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K4A···4H4I4J4K4L6A···6F6G···6X6Y···6AG12A···12P12Q···12AN12AO···12AV
order1222333···33334···444446···66···66···612···1212···1212···12
size1111112···24443···399991···12···24···43···36···69···9

108 irreducible representations

dim111111111122222222444444
type++++-++-+
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6C3xS3C4xS3C3xDic3S3xC6S3xC12S32S3xDic3C6.D6C3xS32C3xS3xDic3C3xC6.D6
kernelC3xDic32Dic3xC3xC6C6xC3:Dic3Dic32C32xDic3C3xC3:Dic3C6xDic3C2xC3:Dic3C3xDic3C3:Dic3C6xDic3C3xDic3C62C2xDic3C3xC6Dic3C2xC6C6C2xC6C6C6C22C2C2
# reps12128442168242488416121242

Matrix representation of C3xDic32 in GL6(F13)

100000
010000
009000
000900
000010
000001
,
100000
010000
001000
000100
000001
0000121
,
1200000
0120000
0012000
0001200
000005
000050
,
110000
1200000
00121200
001000
000010
000001
,
500000
880000
001000
00121200
0000120
0000012

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

C3xDic32 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3^2
% in TeX

G:=Group("C3xDic3^2");
// GroupNames label

G:=SmallGroup(432,425);
// by ID

G=gap.SmallGroup(432,425);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,92,2028,14118]);
// Polycyclic

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

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