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## G = Dic3×SL2(𝔽3)  order 288 = 25·32

### Direct product of Dic3 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — Dic3×SL2(𝔽3)
 Chief series C1 — C2 — C6 — C3×Q8 — C6×Q8 — C6×SL2(𝔽3) — Dic3×SL2(𝔽3)
 Lower central C3×Q8 — Dic3×SL2(𝔽3)
 Upper central C1 — C22

Generators and relations for Dic3×SL2(𝔽3)
G = < a,b,c,d,e | a6=c4=e3=1, b2=a3, d2=c2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 254 in 67 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, Q8, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, SL2(𝔽3), SL2(𝔽3), C2×Dic3, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C4×Q8, C3×Dic3, C62, C4×Dic3, C4⋊Dic3, C2×SL2(𝔽3), C2×SL2(𝔽3), C6×Q8, C3×SL2(𝔽3), C6×Dic3, C4×SL2(𝔽3), Q8×Dic3, C6×SL2(𝔽3), Dic3×SL2(𝔽3)
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, A4, C3×S3, SL2(𝔽3), C2×A4, C3×Dic3, C4×A4, C2×SL2(𝔽3), C4.A4, S3×A4, C4×SL2(𝔽3), Dic3.A4, S3×SL2(𝔽3), Dic3×A4, Dic3×SL2(𝔽3)

Smallest permutation representation of Dic3×SL2(𝔽3)
On 96 points
Generators in S96
(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 56 4 59)(2 55 5 58)(3 60 6 57)(7 75 10 78)(8 74 11 77)(9 73 12 76)(13 62 16 65)(14 61 17 64)(15 66 18 63)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 54 28 51)(26 53 29 50)(27 52 30 49)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
(1 21 41 33)(2 22 42 34)(3 23 37 35)(4 24 38 36)(5 19 39 31)(6 20 40 32)(7 47 49 13)(8 48 50 14)(9 43 51 15)(10 44 52 16)(11 45 53 17)(12 46 54 18)(25 66 73 92)(26 61 74 93)(27 62 75 94)(28 63 76 95)(29 64 77 96)(30 65 78 91)(55 71 87 83)(56 72 88 84)(57 67 89 79)(58 68 90 80)(59 69 85 81)(60 70 86 82)
(1 49 41 7)(2 50 42 8)(3 51 37 9)(4 52 38 10)(5 53 39 11)(6 54 40 12)(13 21 47 33)(14 22 48 34)(15 23 43 35)(16 24 44 36)(17 19 45 31)(18 20 46 32)(25 86 73 60)(26 87 74 55)(27 88 75 56)(28 89 76 57)(29 90 77 58)(30 85 78 59)(61 71 93 83)(62 72 94 84)(63 67 95 79)(64 68 96 80)(65 69 91 81)(66 70 92 82)
(1 3 5)(2 4 6)(7 35 17)(8 36 18)(9 31 13)(10 32 14)(11 33 15)(12 34 16)(19 47 51)(20 48 52)(21 43 53)(22 44 54)(23 45 49)(24 46 50)(25 68 94)(26 69 95)(27 70 96)(28 71 91)(29 72 92)(30 67 93)(37 39 41)(38 40 42)(55 59 57)(56 60 58)(61 78 79)(62 73 80)(63 74 81)(64 75 82)(65 76 83)(66 77 84)(85 89 87)(86 90 88)

G:=sub<Sym(96)| (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,56,4,59)(2,55,5,58)(3,60,6,57)(7,75,10,78)(8,74,11,77)(9,73,12,76)(13,62,16,65)(14,61,17,64)(15,66,18,63)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,54,28,51)(26,53,29,50)(27,52,30,49)(31,80,34,83)(32,79,35,82)(33,84,36,81)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93), (1,21,41,33)(2,22,42,34)(3,23,37,35)(4,24,38,36)(5,19,39,31)(6,20,40,32)(7,47,49,13)(8,48,50,14)(9,43,51,15)(10,44,52,16)(11,45,53,17)(12,46,54,18)(25,66,73,92)(26,61,74,93)(27,62,75,94)(28,63,76,95)(29,64,77,96)(30,65,78,91)(55,71,87,83)(56,72,88,84)(57,67,89,79)(58,68,90,80)(59,69,85,81)(60,70,86,82), (1,49,41,7)(2,50,42,8)(3,51,37,9)(4,52,38,10)(5,53,39,11)(6,54,40,12)(13,21,47,33)(14,22,48,34)(15,23,43,35)(16,24,44,36)(17,19,45,31)(18,20,46,32)(25,86,73,60)(26,87,74,55)(27,88,75,56)(28,89,76,57)(29,90,77,58)(30,85,78,59)(61,71,93,83)(62,72,94,84)(63,67,95,79)(64,68,96,80)(65,69,91,81)(66,70,92,82), (1,3,5)(2,4,6)(7,35,17)(8,36,18)(9,31,13)(10,32,14)(11,33,15)(12,34,16)(19,47,51)(20,48,52)(21,43,53)(22,44,54)(23,45,49)(24,46,50)(25,68,94)(26,69,95)(27,70,96)(28,71,91)(29,72,92)(30,67,93)(37,39,41)(38,40,42)(55,59,57)(56,60,58)(61,78,79)(62,73,80)(63,74,81)(64,75,82)(65,76,83)(66,77,84)(85,89,87)(86,90,88)>;

G:=Group( (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,56,4,59)(2,55,5,58)(3,60,6,57)(7,75,10,78)(8,74,11,77)(9,73,12,76)(13,62,16,65)(14,61,17,64)(15,66,18,63)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,54,28,51)(26,53,29,50)(27,52,30,49)(31,80,34,83)(32,79,35,82)(33,84,36,81)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93), (1,21,41,33)(2,22,42,34)(3,23,37,35)(4,24,38,36)(5,19,39,31)(6,20,40,32)(7,47,49,13)(8,48,50,14)(9,43,51,15)(10,44,52,16)(11,45,53,17)(12,46,54,18)(25,66,73,92)(26,61,74,93)(27,62,75,94)(28,63,76,95)(29,64,77,96)(30,65,78,91)(55,71,87,83)(56,72,88,84)(57,67,89,79)(58,68,90,80)(59,69,85,81)(60,70,86,82), (1,49,41,7)(2,50,42,8)(3,51,37,9)(4,52,38,10)(5,53,39,11)(6,54,40,12)(13,21,47,33)(14,22,48,34)(15,23,43,35)(16,24,44,36)(17,19,45,31)(18,20,46,32)(25,86,73,60)(26,87,74,55)(27,88,75,56)(28,89,76,57)(29,90,77,58)(30,85,78,59)(61,71,93,83)(62,72,94,84)(63,67,95,79)(64,68,96,80)(65,69,91,81)(66,70,92,82), (1,3,5)(2,4,6)(7,35,17)(8,36,18)(9,31,13)(10,32,14)(11,33,15)(12,34,16)(19,47,51)(20,48,52)(21,43,53)(22,44,54)(23,45,49)(24,46,50)(25,68,94)(26,69,95)(27,70,96)(28,71,91)(29,72,92)(30,67,93)(37,39,41)(38,40,42)(55,59,57)(56,60,58)(61,78,79)(62,73,80)(63,74,81)(64,75,82)(65,76,83)(66,77,84)(85,89,87)(86,90,88) );

G=PermutationGroup([[(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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,56,4,59),(2,55,5,58),(3,60,6,57),(7,75,10,78),(8,74,11,77),(9,73,12,76),(13,62,16,65),(14,61,17,64),(15,66,18,63),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,54,28,51),(26,53,29,50),(27,52,30,49),(31,80,34,83),(32,79,35,82),(33,84,36,81),(37,86,40,89),(38,85,41,88),(39,90,42,87),(43,92,46,95),(44,91,47,94),(45,96,48,93)], [(1,21,41,33),(2,22,42,34),(3,23,37,35),(4,24,38,36),(5,19,39,31),(6,20,40,32),(7,47,49,13),(8,48,50,14),(9,43,51,15),(10,44,52,16),(11,45,53,17),(12,46,54,18),(25,66,73,92),(26,61,74,93),(27,62,75,94),(28,63,76,95),(29,64,77,96),(30,65,78,91),(55,71,87,83),(56,72,88,84),(57,67,89,79),(58,68,90,80),(59,69,85,81),(60,70,86,82)], [(1,49,41,7),(2,50,42,8),(3,51,37,9),(4,52,38,10),(5,53,39,11),(6,54,40,12),(13,21,47,33),(14,22,48,34),(15,23,43,35),(16,24,44,36),(17,19,45,31),(18,20,46,32),(25,86,73,60),(26,87,74,55),(27,88,75,56),(28,89,76,57),(29,90,77,58),(30,85,78,59),(61,71,93,83),(62,72,94,84),(63,67,95,79),(64,68,96,80),(65,69,91,81),(66,70,92,82)], [(1,3,5),(2,4,6),(7,35,17),(8,36,18),(9,31,13),(10,32,14),(11,33,15),(12,34,16),(19,47,51),(20,48,52),(21,43,53),(22,44,54),(23,45,49),(24,46,50),(25,68,94),(26,69,95),(27,70,96),(28,71,91),(29,72,92),(30,67,93),(37,39,41),(38,40,42),(55,59,57),(56,60,58),(61,78,79),(62,73,80),(63,74,81),(64,75,82),(65,76,83),(66,77,84),(85,89,87),(86,90,88)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D ··· 6I 6J ··· 6O 12A ··· 12J order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 4 4 8 8 3 3 3 3 6 6 18 18 2 2 2 4 ··· 4 8 ··· 8 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 6 6 type + + + - - + + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 SL2(𝔽3) SL2(𝔽3) C3×Dic3 C4.A4 A4 C2×A4 C4×A4 Dic3.A4 Dic3.A4 S3×SL2(𝔽3) S3×SL2(𝔽3) S3×A4 Dic3×A4 kernel Dic3×SL2(𝔽3) C6×SL2(𝔽3) Q8×Dic3 C3×SL2(𝔽3) C6×Q8 C3×Q8 C2×SL2(𝔽3) SL2(𝔽3) C2×Q8 Dic3 Dic3 Q8 C6 C2×Dic3 C2×C6 C6 C2 C2 C2 C2 C22 C2 # reps 1 1 2 2 2 4 1 1 2 2 4 2 6 1 1 2 1 2 1 2 1 1

Matrix representation of Dic3×SL2(𝔽3) in GL4(𝔽13) generated by

 4 0 0 0 9 10 0 0 0 0 1 0 0 0 0 1
,
 12 8 0 0 3 1 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 9 3 0 0 3 4
,
 3 0 0 0 0 3 0 0 0 0 0 4 0 0 1 10
G:=sub<GL(4,GF(13))| [4,9,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[12,3,0,0,8,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4],[3,0,0,0,0,3,0,0,0,0,0,1,0,0,4,10] >;

Dic3×SL2(𝔽3) in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("Dic3xSL(2,3)");
// GroupNames label

G:=SmallGroup(288,409);
// by ID

G=gap.SmallGroup(288,409);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,-2,42,514,360,221,515,242,4037]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^4=e^3=1,b^2=a^3,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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