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## G = SL2(𝔽3).Dic3order 288 = 25·32

### The non-split extension by SL2(𝔽3) of Dic3 acting through Inn(SL2(𝔽3))

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — SL2(𝔽3).Dic3
 Chief series C1 — C2 — C6 — C3×Q8 — C3×C4○D4 — C3×C4.A4 — SL2(𝔽3).Dic3
 Lower central C3×Q8 — SL2(𝔽3).Dic3
 Upper central C1 — C4

Generators and relations for SL2(𝔽3).Dic3
G = < a,b,c,d,e | a4=c3=1, b2=d6=a2, e2=a2d3, bab-1=a-1, cac-1=b, ad=da, ae=ea, cbc-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 24A ··· 24H order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 6 8 8 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 6 2 4 4 8 8 1 1 6 2 4 4 8 8 12 3 3 3 3 18 18 2 2 4 4 4 4 8 8 8 8 12 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 6 6 type + + + - + + + - image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 C8.A4 A4 C2×A4 C4×A4 SL2(𝔽3).Dic3 S3×A4 Dic3×A4 kernel SL2(𝔽3).Dic3 C3×C4.A4 D4.Dic3 C3×SL2(𝔽3) C3×C4○D4 C3×Q8 C4.A4 SL2(𝔽3) C4○D4 Q8 C3 C3⋊C8 C12 C6 C1 C4 C2 # reps 1 1 2 2 2 4 1 1 2 2 12 1 1 2 6 1 1

Matrix representation of SL2(𝔽3).Dic3 in GL4(𝔽5) generated by

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

SL2(𝔽3).Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("SL(2,3).Dic3");
// GroupNames label

G:=SmallGroup(288,410);
// by ID

G=gap.SmallGroup(288,410);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,-2,42,520,514,360,221,515,242,4037]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=1,b^2=d^6=a^2,e^2=a^2*d^3,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations

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