Copied to
clipboard

## G = C3×Dic3.A4order 432 = 24·33

### Direct product of C3 and Dic3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C3×Dic3.A4
 Chief series C1 — C2 — C6 — C3×Q8 — Q8×C32 — C32×SL2(𝔽3) — C3×Dic3.A4
 Lower central C3×Q8 — C3×Dic3.A4
 Upper central C1 — C6

Generators and relations for C3×Dic3.A4
G = < a,b,c,d,e,f | a3=b6=f3=1, c2=d2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b3d, fdf-1=b3de, fef-1=d >

Subgroups: 450 in 105 conjugacy classes, 28 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, SL2(𝔽3), SL2(𝔽3), C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C33, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C4.A4, Q83S3, C3×C4○D4, C32×C6, C3×SL2(𝔽3), C3×SL2(𝔽3), C3×SL2(𝔽3), S3×C12, C3×D12, Q8×C32, C32×Dic3, Dic3.A4, C3×C4.A4, C3×Q83S3, C32×SL2(𝔽3), C3×Dic3.A4
Quotients: C1, C2, C3, S3, C6, C32, A4, C3×S3, C3×C6, C2×A4, C3×A4, C4.A4, S3×C32, S3×A4, C6×A4, Dic3.A4, C3×C4.A4, C3×S3×A4, C3×Dic3.A4

Smallest permutation representation of C3×Dic3.A4
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(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 26 4 29)(2 25 5 28)(3 30 6 27)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 13 4 16)(2 14 5 17)(3 15 6 18)(7 19 10 22)(8 20 11 23)(9 21 12 24)(25 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 48)
(1 7 4 10)(2 8 5 11)(3 9 6 12)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 31 28 34)(26 32 29 35)(27 33 30 36)(37 46 40 43)(38 47 41 44)(39 48 42 45)
(7 19 13)(8 20 14)(9 21 15)(10 22 16)(11 23 17)(12 24 18)(31 43 37)(32 44 38)(33 45 39)(34 46 40)(35 47 41)(36 48 42)

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

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

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

63 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3Q 4A 4B 4C 6A 6B 6C 6D 6E 6F ··· 6K 6L ··· 6Q 6R 6S 12A 12B 12C 12D 12E 12F 12G ··· 12U order 1 2 2 3 3 3 3 3 3 ··· 3 3 ··· 3 4 4 4 6 6 6 6 6 6 ··· 6 6 ··· 6 6 6 12 12 12 12 12 12 12 ··· 12 size 1 1 18 1 1 2 2 2 4 ··· 4 8 ··· 8 3 3 6 1 1 2 2 2 4 ··· 4 8 ··· 8 18 18 3 3 3 3 6 6 12 ··· 12

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 6 6 type + + + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 C4.A4 C3×C4.A4 A4 C2×A4 C3×A4 C6×A4 Dic3.A4 Dic3.A4 C3×Dic3.A4 S3×A4 C3×S3×A4 kernel C3×Dic3.A4 C32×SL2(𝔽3) Dic3.A4 C3×Q8⋊3S3 C3×SL2(𝔽3) Q8×C32 C3×SL2(𝔽3) SL2(𝔽3) C3×Q8 C32 C3 C3×Dic3 C3×C6 Dic3 C6 C3 C3 C1 C6 C2 # reps 1 1 6 2 6 2 1 6 2 6 12 1 1 2 2 1 2 6 1 2

Matrix representation of C3×Dic3.A4 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 3 0 0 0 6 6 1 1 0 5 3 5 1 6 6 4
,
 2 5 5 2 3 3 1 5 0 1 5 1 4 6 1 4
,
 4 1 1 1 3 0 3 6 6 5 1 2 2 5 6 2
,
 4 2 2 2 5 4 6 5 3 6 2 1 1 0 2 4
,
 1 0 0 0 0 6 2 5 0 0 4 0 0 5 5 6
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[3,6,0,1,0,6,5,6,0,1,3,6,0,1,5,4],[2,3,0,4,5,3,1,6,5,1,5,1,2,5,1,4],[4,3,6,2,1,0,5,5,1,3,1,6,1,6,2,2],[4,5,3,1,2,4,6,0,2,6,2,2,2,5,1,4],[1,0,0,0,0,6,0,5,0,2,4,5,0,5,0,6] >;

C3×Dic3.A4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3.A_4
% in TeX

G:=Group("C3xDic3.A4");
// GroupNames label

G:=SmallGroup(432,622);
// by ID

G=gap.SmallGroup(432,622);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-3,-2,1512,766,360,326,515,242,6053]);
// Polycyclic

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

׿
×
𝔽