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G = C3xDic12order 144 = 24·32

Direct product of C3 and Dic12

direct product, metacyclic, supersoluble, monomial

Aliases: C3xDic12, C24.1C6, C24.5S3, C32:4Q16, C12.62D6, C6.21D12, Dic6.1C6, C8.(C3xS3), C3:1(C3xQ16), C6.3(C3xD4), C4.10(S3xC6), (C3xC24).2C2, (C3xC6).19D4, C2.5(C3xD12), C12.10(C2xC6), (C3xDic6).4C2, (C3xC12).39C22, SmallGroup(144,73)

Series: Derived Chief Lower central Upper central

C1C12 — C3xDic12
C1C3C6C12C3xC12C3xDic6 — C3xDic12
C3C6C12 — C3xDic12
C1C6C12C24

Generators and relations for C3xDic12
 G = < a,b,c | a3=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 80 in 40 conjugacy classes, 22 normal (18 characteristic)
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, Q16, C3xS3, D12, C3xD4, S3xC6, Dic12, C3xQ16, C3xD12, C3xDic12
2C3
6C4
6C4
2C6
3Q8
3Q8
2C12
2Dic3
2Dic3
6C12
6C12
3Q16
2C24
3C3xQ8
3C3xQ8
2C3xDic3
2C3xDic3
3C3xQ16

Smallest permutation representation of C3xDic12
On 48 points
Generators in S48
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 37 13 25)(2 36 14 48)(3 35 15 47)(4 34 16 46)(5 33 17 45)(6 32 18 44)(7 31 19 43)(8 30 20 42)(9 29 21 41)(10 28 22 40)(11 27 23 39)(12 26 24 38)

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

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

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

C3xDic12 is a maximal subgroup of
D24.S3  C24.49D6  C32:2Q32  C32:3Q32  Dic12:S3  C24.23D6  D6.3D12  D24:5S3  D12.4D6  C3xS3xQ16  He3:4Q16  C72.C6  He3:5Q16
C3xDic12 is a maximal quotient of
He3:4Q16  C72.C6

45 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B12A···12H12I12J12K12L24A···24P
order1233333444666668812···121212121224···24
size11112222121211222222···2121212122···2

45 irreducible representations

dim111111222222222222
type++++++-+-
imageC1C2C2C3C6C6S3D4D6Q16C3xS3D12C3xD4S3xC6Dic12C3xQ16C3xD12C3xDic12
kernelC3xDic12C3xC24C3xDic6Dic12C24Dic6C24C3xC6C12C32C8C6C6C4C3C3C2C1
# reps112224111222224448

Matrix representation of C3xDic12 in GL2(F73) generated by

640
064
,
430
017
,
01
720
G:=sub<GL(2,GF(73))| [64,0,0,64],[43,0,0,17],[0,72,1,0] >;

C3xDic12 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{12}
% in TeX

G:=Group("C3xDic12");
// GroupNames label

G:=SmallGroup(144,73);
// by ID

G=gap.SmallGroup(144,73);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,223,867,69,3461]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3xDic12 in TeX

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