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## G = C3×C6.6S4order 432 = 24·33

### Direct product of C3 and C6.6S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C3×C6.6S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C32×SL2(𝔽3) — C3×C6.6S4
 Lower central C3×SL2(𝔽3) — C3×C6.6S4
 Upper central C1 — C6

Generators and relations for C3×C6.6S4
G = < a,b,c,d,e,f | a3=b6=e3=f2=1, c2=d2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b3c, ece-1=b3cd, fcf=cd, ede-1=c, fdf=b3d, fef=e-1 >

Subgroups: 710 in 117 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), D12, C3×D4, C3×Q8, C3×Q8, C33, C3×C12, S3×C6, C2×C3⋊S3, Q82S3, C3×SD16, GL2(𝔽3), C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×SL2(𝔽3), C3×SL2(𝔽3), C3×SL2(𝔽3), C3×D12, Q8×C32, C6×C3⋊S3, C3×Q82S3, C3×GL2(𝔽3), C6.6S4, C32×SL2(𝔽3), C3×C6.6S4
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, S4, GL2(𝔽3), C3×C3⋊S3, C3×S4, C3⋊S4, C3×GL2(𝔽3), C6.6S4, C3×C3⋊S4, C3×C6.6S4

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 3F ··· 3N 4 6A 6B 6C 6D 6E 6F ··· 6N 6O 6P 8A 8B 12A 12B 12C 12D 12E 24A 24B 24C 24D order 1 2 2 3 3 3 3 3 3 ··· 3 4 6 6 6 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 24 24 24 24 size 1 1 36 1 1 2 2 2 8 ··· 8 6 1 1 2 2 2 8 ··· 8 36 36 18 18 6 6 12 12 12 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 GL2(𝔽3) C3×GL2(𝔽3) S4 C3×S4 GL2(𝔽3) C3×GL2(𝔽3) C6.6S4 C3×C6.6S4 C3⋊S4 C3×C3⋊S4 kernel C3×C6.6S4 C32×SL2(𝔽3) C6.6S4 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×C6.6S4 in GL4(𝔽7) generated by

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

C3×C6.6S4 in GAP, Magma, Sage, TeX

C_3\times C_6._6S_4
% in TeX

G:=Group("C3xC6.6S4");
// GroupNames label

G:=SmallGroup(432,617);
// by ID

G=gap.SmallGroup(432,617);
# by ID

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

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

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