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## G = C3⋊S3×SL2(𝔽3)  order 432 = 24·33

### Direct product of C3⋊S3 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C32 — C3⋊S3×SL2(𝔽3)
 Chief series C1 — C2 — C6 — C3×C6 — Q8×C32 — C32×SL2(𝔽3) — C3⋊S3×SL2(𝔽3)
 Lower central Q8×C32 — C3⋊S3×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for C3⋊S3×SL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c2=d4=f3=1, e2=d2, ab=ba, cac=a-1, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=d-1, fdf-1=e, fef-1=de >

Subgroups: 806 in 132 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C3×Q8, C33, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×SL2(𝔽3), S3×Q8, C3×C3⋊S3, C32×C6, C3×SL2(𝔽3), C3×SL2(𝔽3), C324Q8, C4×C3⋊S3, Q8×C32, C6×C3⋊S3, S3×SL2(𝔽3), Q8×C3⋊S3, C32×SL2(𝔽3), C3⋊S3×SL2(𝔽3)
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C3⋊S3, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C3×C3⋊S3, S3×A4, S3×SL2(𝔽3), A4×C3⋊S3, C3⋊S3×SL2(𝔽3)

Smallest permutation representation of C3⋊S3×SL2(𝔽3)
On 72 points
Generators in S72
(1 13 5)(2 14 6)(3 15 7)(4 16 8)(9 36 37)(10 33 38)(11 34 39)(12 35 40)(17 62 69)(18 63 70)(19 64 71)(20 61 72)(21 25 29)(22 26 30)(23 27 31)(24 28 32)(41 52 67)(42 49 68)(43 50 65)(44 51 66)(45 56 60)(46 53 57)(47 54 58)(48 55 59)
(1 23 35)(2 24 36)(3 21 33)(4 22 34)(5 31 12)(6 32 9)(7 29 10)(8 30 11)(13 27 40)(14 28 37)(15 25 38)(16 26 39)(17 65 47)(18 66 48)(19 67 45)(20 68 46)(41 56 64)(42 53 61)(43 54 62)(44 55 63)(49 57 72)(50 58 69)(51 59 70)(52 60 71)
(5 13)(6 14)(7 15)(8 16)(9 28)(10 25)(11 26)(12 27)(17 54)(18 55)(19 56)(20 53)(21 33)(22 34)(23 35)(24 36)(29 38)(30 39)(31 40)(32 37)(41 67)(42 68)(43 65)(44 66)(45 64)(46 61)(47 62)(48 63)(57 72)(58 69)(59 70)(60 71)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)
(1 49 3 51)(2 52 4 50)(5 42 7 44)(6 41 8 43)(9 64 11 62)(10 63 12 61)(13 68 15 66)(14 67 16 65)(17 37 19 39)(18 40 20 38)(21 59 23 57)(22 58 24 60)(25 48 27 46)(26 47 28 45)(29 55 31 53)(30 54 32 56)(33 70 35 72)(34 69 36 71)
(2 52 49)(4 50 51)(6 41 42)(8 43 44)(9 64 61)(11 62 63)(14 67 68)(16 65 66)(17 18 39)(19 20 37)(22 58 59)(24 60 57)(26 47 48)(28 45 46)(30 54 55)(32 56 53)(34 69 70)(36 71 72)

G:=sub<Sym(72)| (1,13,5)(2,14,6)(3,15,7)(4,16,8)(9,36,37)(10,33,38)(11,34,39)(12,35,40)(17,62,69)(18,63,70)(19,64,71)(20,61,72)(21,25,29)(22,26,30)(23,27,31)(24,28,32)(41,52,67)(42,49,68)(43,50,65)(44,51,66)(45,56,60)(46,53,57)(47,54,58)(48,55,59), (1,23,35)(2,24,36)(3,21,33)(4,22,34)(5,31,12)(6,32,9)(7,29,10)(8,30,11)(13,27,40)(14,28,37)(15,25,38)(16,26,39)(17,65,47)(18,66,48)(19,67,45)(20,68,46)(41,56,64)(42,53,61)(43,54,62)(44,55,63)(49,57,72)(50,58,69)(51,59,70)(52,60,71), (5,13)(6,14)(7,15)(8,16)(9,28)(10,25)(11,26)(12,27)(17,54)(18,55)(19,56)(20,53)(21,33)(22,34)(23,35)(24,36)(29,38)(30,39)(31,40)(32,37)(41,67)(42,68)(43,65)(44,66)(45,64)(46,61)(47,62)(48,63)(57,72)(58,69)(59,70)(60,71), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,49,3,51)(2,52,4,50)(5,42,7,44)(6,41,8,43)(9,64,11,62)(10,63,12,61)(13,68,15,66)(14,67,16,65)(17,37,19,39)(18,40,20,38)(21,59,23,57)(22,58,24,60)(25,48,27,46)(26,47,28,45)(29,55,31,53)(30,54,32,56)(33,70,35,72)(34,69,36,71), (2,52,49)(4,50,51)(6,41,42)(8,43,44)(9,64,61)(11,62,63)(14,67,68)(16,65,66)(17,18,39)(19,20,37)(22,58,59)(24,60,57)(26,47,48)(28,45,46)(30,54,55)(32,56,53)(34,69,70)(36,71,72)>;

G:=Group( (1,13,5)(2,14,6)(3,15,7)(4,16,8)(9,36,37)(10,33,38)(11,34,39)(12,35,40)(17,62,69)(18,63,70)(19,64,71)(20,61,72)(21,25,29)(22,26,30)(23,27,31)(24,28,32)(41,52,67)(42,49,68)(43,50,65)(44,51,66)(45,56,60)(46,53,57)(47,54,58)(48,55,59), (1,23,35)(2,24,36)(3,21,33)(4,22,34)(5,31,12)(6,32,9)(7,29,10)(8,30,11)(13,27,40)(14,28,37)(15,25,38)(16,26,39)(17,65,47)(18,66,48)(19,67,45)(20,68,46)(41,56,64)(42,53,61)(43,54,62)(44,55,63)(49,57,72)(50,58,69)(51,59,70)(52,60,71), (5,13)(6,14)(7,15)(8,16)(9,28)(10,25)(11,26)(12,27)(17,54)(18,55)(19,56)(20,53)(21,33)(22,34)(23,35)(24,36)(29,38)(30,39)(31,40)(32,37)(41,67)(42,68)(43,65)(44,66)(45,64)(46,61)(47,62)(48,63)(57,72)(58,69)(59,70)(60,71), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,49,3,51)(2,52,4,50)(5,42,7,44)(6,41,8,43)(9,64,11,62)(10,63,12,61)(13,68,15,66)(14,67,16,65)(17,37,19,39)(18,40,20,38)(21,59,23,57)(22,58,24,60)(25,48,27,46)(26,47,28,45)(29,55,31,53)(30,54,32,56)(33,70,35,72)(34,69,36,71), (2,52,49)(4,50,51)(6,41,42)(8,43,44)(9,64,61)(11,62,63)(14,67,68)(16,65,66)(17,18,39)(19,20,37)(22,58,59)(24,60,57)(26,47,48)(28,45,46)(30,54,55)(32,56,53)(34,69,70)(36,71,72) );

G=PermutationGroup([[(1,13,5),(2,14,6),(3,15,7),(4,16,8),(9,36,37),(10,33,38),(11,34,39),(12,35,40),(17,62,69),(18,63,70),(19,64,71),(20,61,72),(21,25,29),(22,26,30),(23,27,31),(24,28,32),(41,52,67),(42,49,68),(43,50,65),(44,51,66),(45,56,60),(46,53,57),(47,54,58),(48,55,59)], [(1,23,35),(2,24,36),(3,21,33),(4,22,34),(5,31,12),(6,32,9),(7,29,10),(8,30,11),(13,27,40),(14,28,37),(15,25,38),(16,26,39),(17,65,47),(18,66,48),(19,67,45),(20,68,46),(41,56,64),(42,53,61),(43,54,62),(44,55,63),(49,57,72),(50,58,69),(51,59,70),(52,60,71)], [(5,13),(6,14),(7,15),(8,16),(9,28),(10,25),(11,26),(12,27),(17,54),(18,55),(19,56),(20,53),(21,33),(22,34),(23,35),(24,36),(29,38),(30,39),(31,40),(32,37),(41,67),(42,68),(43,65),(44,66),(45,64),(46,61),(47,62),(48,63),(57,72),(58,69),(59,70),(60,71)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)], [(1,49,3,51),(2,52,4,50),(5,42,7,44),(6,41,8,43),(9,64,11,62),(10,63,12,61),(13,68,15,66),(14,67,16,65),(17,37,19,39),(18,40,20,38),(21,59,23,57),(22,58,24,60),(25,48,27,46),(26,47,28,45),(29,55,31,53),(30,54,32,56),(33,70,35,72),(34,69,36,71)], [(2,52,49),(4,50,51),(6,41,42),(8,43,44),(9,64,61),(11,62,63),(14,67,68),(16,65,66),(17,18,39),(19,20,37),(22,58,59),(24,60,57),(26,47,48),(28,45,46),(30,54,55),(32,56,53),(34,69,70),(36,71,72)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G ··· 3N 4A 4B 6A 6B 6C 6D 6E 6F 6G ··· 6N 6O 6P 6Q 6R 12A 12B 12C 12D order 1 2 2 2 3 3 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 9 9 2 2 2 2 4 4 8 ··· 8 6 54 2 2 2 2 4 4 8 ··· 8 36 36 36 36 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 4 4 6 type + + + - + + - + image C1 C2 C3 C6 S3 C3×S3 SL2(𝔽3) SL2(𝔽3) A4 C2×A4 S3×SL2(𝔽3) S3×SL2(𝔽3) S3×A4 kernel C3⋊S3×SL2(𝔽3) C32×SL2(𝔽3) Q8×C3⋊S3 Q8×C32 C3×SL2(𝔽3) C3×Q8 C3⋊S3 C3⋊S3 C2×C3⋊S3 C3×C6 C3 C3 C6 # reps 1 1 2 2 4 8 2 4 1 1 4 8 4

Matrix representation of C3⋊S3×SL2(𝔽3) in GL6(𝔽13)

 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 3 0 0 0 0 3 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 9 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,9,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3⋊S3×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3:S3xSL(2,3)");
// GroupNames label

G:=SmallGroup(432,626);
// by ID

G=gap.SmallGroup(432,626);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-3,-2,198,772,94,1081,528,1684,6053]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^4=f^3=1,e^2=d^2,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=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=d^-1,f*d*f^-1=e,f*e*f^-1=d*e>;
// generators/relations

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