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G = C3×Dic12order 144 = 24·32

Direct product of C3 and Dic12

direct product, metacyclic, supersoluble, monomial

Aliases: C3×Dic12, C24.1C6, C24.5S3, C324Q16, C12.62D6, C6.21D12, Dic6.1C6, C8.(C3×S3), C31(C3×Q16), C6.3(C3×D4), C4.10(S3×C6), (C3×C24).2C2, (C3×C6).19D4, C2.5(C3×D12), C12.10(C2×C6), (C3×Dic6).4C2, (C3×C12).39C22, SmallGroup(144,73)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Dic12
C1C3C6C12C3×C12C3×Dic6 — C3×Dic12
C3C6C12 — C3×Dic12
C1C6C12C24

Generators and relations for C3×Dic12
 G = < a,b,c | a3=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

2C3
6C4
6C4
2C6
3Q8
3Q8
2C12
2Dic3
2Dic3
6C12
6C12
3Q16
2C24
3C3×Q8
3C3×Q8
2C3×Dic3
2C3×Dic3
3C3×Q16

Smallest permutation representation of C3×Dic12
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 25 13 37)(2 48 14 36)(3 47 15 35)(4 46 16 34)(5 45 17 33)(6 44 18 32)(7 43 19 31)(8 42 20 30)(9 41 21 29)(10 40 22 28)(11 39 23 27)(12 38 24 26)

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

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

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

C3×Dic12 is a maximal subgroup of
D24.S3  C24.49D6  C322Q32  C323Q32  Dic12⋊S3  C24.23D6  D6.3D12  D245S3  D12.4D6  C3×S3×Q16  He34Q16  C72.C6  He35Q16
C3×Dic12 is a maximal quotient of
He34Q16  C72.C6

45 conjugacy classes

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

45 irreducible representations

dim111111222222222222
type++++++-+-
imageC1C2C2C3C6C6S3D4D6Q16C3×S3D12C3×D4S3×C6Dic12C3×Q16C3×D12C3×Dic12
kernelC3×Dic12C3×C24C3×Dic6Dic12C24Dic6C24C3×C6C12C32C8C6C6C4C3C3C2C1
# reps112224111222224448

Matrix representation of C3×Dic12 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] >;

C3×Dic12 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 C3×Dic12 in TeX

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