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## G = C2×C6×SL2(𝔽3)  order 288 = 25·32

### Direct product of C2×C6 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C6×SL2(𝔽3)
G = < a,b,c,d,e | a2=b6=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 432 in 164 conjugacy classes, 62 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, Q8, C23, C32, C12, C2×C6, C2×C6, C22×C4, C2×Q8, C2×Q8, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, C22×C6, C22×C6, C22×Q8, C62, C2×SL2(𝔽3), C22×C12, C6×Q8, C6×Q8, C3×SL2(𝔽3), C2×C62, C22×SL2(𝔽3), Q8×C2×C6, C6×SL2(𝔽3), C2×C6×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, SL2(𝔽3), C2×A4, C3×A4, C62, C2×SL2(𝔽3), C22×A4, C3×SL2(𝔽3), C6×A4, C22×SL2(𝔽3), C6×SL2(𝔽3), A4×C2×C6, C2×C6×SL2(𝔽3)

Smallest permutation representation of C2×C6×SL2(𝔽3)
On 96 points
Generators in S96
(1 42)(2 37)(3 38)(4 39)(5 40)(6 41)(7 74)(8 75)(9 76)(10 77)(11 78)(12 73)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 54)(26 49)(27 50)(28 51)(29 52)(30 53)(31 48)(32 43)(33 44)(34 45)(35 46)(36 47)(55 83)(56 84)(57 79)(58 80)(59 81)(60 82)(61 70)(62 71)(63 72)(64 67)(65 68)(66 69)(85 95)(86 96)(87 91)(88 92)(89 93)(90 94)
(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 45 81 89)(2 46 82 90)(3 47 83 85)(4 48 84 86)(5 43 79 87)(6 44 80 88)(7 17 25 63)(8 18 26 64)(9 13 27 65)(10 14 28 66)(11 15 29 61)(12 16 30 62)(19 53 71 73)(20 54 72 74)(21 49 67 75)(22 50 68 76)(23 51 69 77)(24 52 70 78)(31 56 96 39)(32 57 91 40)(33 58 92 41)(34 59 93 42)(35 60 94 37)(36 55 95 38)
(1 19 81 71)(2 20 82 72)(3 21 83 67)(4 22 84 68)(5 23 79 69)(6 24 80 70)(7 94 25 35)(8 95 26 36)(9 96 27 31)(10 91 28 32)(11 92 29 33)(12 93 30 34)(13 56 65 39)(14 57 66 40)(15 58 61 41)(16 59 62 42)(17 60 63 37)(18 55 64 38)(43 77 87 51)(44 78 88 52)(45 73 89 53)(46 74 90 54)(47 75 85 49)(48 76 86 50)
(7 17 35)(8 18 36)(9 13 31)(10 14 32)(11 15 33)(12 16 34)(19 45 73)(20 46 74)(21 47 75)(22 48 76)(23 43 77)(24 44 78)(25 63 94)(26 64 95)(27 65 96)(28 66 91)(29 61 92)(30 62 93)(49 67 85)(50 68 86)(51 69 87)(52 70 88)(53 71 89)(54 72 90)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A ··· 6N 6O ··· 6BD 12A ··· 12H order 1 2 ··· 2 3 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 4 ··· 4 6 6 6 6 1 ··· 1 4 ··· 4 6 ··· 6

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 type + + - + + image C1 C2 C3 C3 C6 C6 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) A4 C2×A4 C3×A4 C6×A4 kernel C2×C6×SL2(𝔽3) C6×SL2(𝔽3) C22×SL2(𝔽3) Q8×C2×C6 C2×SL2(𝔽3) C6×Q8 C2×C6 C2×C6 C22 C22×C6 C2×C6 C23 C22 # reps 1 3 6 2 18 6 4 8 24 1 3 2 6

Matrix representation of C2×C6×SL2(𝔽3) in GL4(𝔽13) generated by

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

C2×C6×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times C_6\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xC6xSL(2,3)");
// GroupNames label

G:=SmallGroup(288,981);
// by ID

G=gap.SmallGroup(288,981);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,648,172,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^4=e^3=1,d^2=c^2,a*b=b*a,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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