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## G = C3×C6.5S4order 432 = 24·33

### Direct product of C3 and C6.5S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C3×C6.5S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C32×SL2(𝔽3) — C3×C6.5S4
 Lower central C3×SL2(𝔽3) — C3×C6.5S4
 Upper central C1 — C6

Generators and relations for C3×C6.5S4
G = < a,b,c,d,e,f | a3=b6=e3=1, c2=d2=f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=b3c, ece-1=b3cd, fcf-1=cd, ede-1=c, fdf-1=b3d, fef-1=e-1 >

Subgroups: 454 in 97 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C3×Q8, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3⋊Q16, C3×Q16, CSU2(𝔽3), C32×C6, C3×C3⋊C8, C3×SL2(𝔽3), C3×SL2(𝔽3), C3×SL2(𝔽3), C3×Dic6, Q8×C32, C3×C3⋊Dic3, C3×C3⋊Q16, C3×CSU2(𝔽3), C6.5S4, C32×SL2(𝔽3), C3×C6.5S4
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, S4, CSU2(𝔽3), C3×C3⋊S3, C3×S4, C3⋊S4, C3×CSU2(𝔽3), C6.5S4, C3×C3⋊S4, C3×C6.5S4

Smallest permutation representation of C3×C6.5S4
On 48 points
Generators in S48
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(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 23 4 20)(2 24 5 21)(3 19 6 22)(7 37 10 40)(8 38 11 41)(9 39 12 42)(13 48 16 45)(14 43 17 46)(15 44 18 47)(25 34 28 31)(26 35 29 32)(27 36 30 33)
(1 32 4 35)(2 33 5 36)(3 34 6 31)(7 15 10 18)(8 16 11 13)(9 17 12 14)(19 25 22 28)(20 26 23 29)(21 27 24 30)(37 47 40 44)(38 48 41 45)(39 43 42 46)
(7 37 47)(8 38 48)(9 39 43)(10 40 44)(11 41 45)(12 42 46)(19 34 28)(20 35 29)(21 36 30)(22 31 25)(23 32 26)(24 33 27)
(1 14 4 17)(2 13 5 16)(3 18 6 15)(7 34 10 31)(8 33 11 36)(9 32 12 35)(19 40 22 37)(20 39 23 42)(21 38 24 41)(25 47 28 44)(26 46 29 43)(27 45 30 48)

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

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

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

45 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3N 4A 4B 6A 6B 6C 6D 6E 6F ··· 6N 8A 8B 12A 12B 12C 12D 12E 12F 12G 24A 24B 24C 24D order 1 2 3 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 6 ··· 6 8 8 12 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 2 2 2 8 ··· 8 6 36 1 1 2 2 2 8 ··· 8 18 18 6 6 12 12 12 36 36 18 18 18 18

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 4 4 4 4 6 6 type + + + + - + - - + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 CSU2(𝔽3) C3×CSU2(𝔽3) S4 C3×S4 CSU2(𝔽3) C3×CSU2(𝔽3) C6.5S4 C3×C6.5S4 C3⋊S4 C3×C3⋊S4 kernel C3×C6.5S4 C32×SL2(𝔽3) C6.5S4 C3×SL2(𝔽3) C3×SL2(𝔽3) Q8×C32 SL2(𝔽3) C3×Q8 C32 C3 C3×C6 C6 C32 C3 C3 C1 C6 C2 # reps 1 1 2 2 3 1 6 2 2 4 2 4 1 2 3 6 1 2

Matrix representation of C3×C6.5S4 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
,
 3 6 6 6 4 0 4 1 1 2 6 5 5 2 1 5
,
 4 2 2 2 5 4 6 5 3 6 2 1 1 0 2 4
,
 5 5 5 5 2 6 0 3 4 1 2 6 6 1 6 6
,
 0 6 0 6 3 0 4 4 2 4 4 3 5 0 3 3
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],[3,4,1,5,6,0,2,2,6,4,6,1,6,1,5,5],[4,5,3,1,2,4,6,0,2,6,2,2,2,5,1,4],[5,2,4,6,5,6,1,1,5,0,2,6,5,3,6,6],[0,3,2,5,6,0,4,0,0,4,4,3,6,4,3,3] >;

C3×C6.5S4 in GAP, Magma, Sage, TeX

C_3\times C_6._5S_4
% in TeX

G:=Group("C3xC6.5S4");
// GroupNames label

G:=SmallGroup(432,616);
// by ID

G=gap.SmallGroup(432,616);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,254,1011,3784,1908,172,2273,1153,285,124]);
// Polycyclic

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

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