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## G = C12×SL2(𝔽3)  order 288 = 25·32

### Direct product of C12 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C12×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C2×Q8 — C6×Q8 — C6×SL2(𝔽3) — C12×SL2(𝔽3)
 Lower central Q8 — C12×SL2(𝔽3)
 Upper central C1 — C2×C12

Generators and relations for C12×SL2(𝔽3)
G = < a,b,c,d | a12=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Subgroups: 216 in 82 conjugacy classes, 34 normal (21 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, Q8, C32, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, SL2(𝔽3), C2×C12, C2×C12, C3×Q8, C3×Q8, C4×Q8, C3×C12, C62, C4×C12, C3×C4⋊C4, C2×SL2(𝔽3), C6×Q8, C3×SL2(𝔽3), C6×C12, C4×SL2(𝔽3), Q8×C12, C6×SL2(𝔽3), C12×SL2(𝔽3)
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3×C6, SL2(𝔽3), C2×A4, C3×C12, C3×A4, C4×A4, C2×SL2(𝔽3), C4.A4, C3×SL2(𝔽3), C6×A4, C4×SL2(𝔽3), C12×A4, C6×SL2(𝔽3), C3×C4.A4, C12×SL2(𝔽3)

Smallest permutation representation of C12×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 78 46)(2 57 79 47)(3 58 80 48)(4 59 81 37)(5 60 82 38)(6 49 83 39)(7 50 84 40)(8 51 73 41)(9 52 74 42)(10 53 75 43)(11 54 76 44)(12 55 77 45)(13 66 33 93)(14 67 34 94)(15 68 35 95)(16 69 36 96)(17 70 25 85)(18 71 26 86)(19 72 27 87)(20 61 28 88)(21 62 29 89)(22 63 30 90)(23 64 31 91)(24 65 32 92)
(1 28 78 20)(2 29 79 21)(3 30 80 22)(4 31 81 23)(5 32 82 24)(6 33 83 13)(7 34 84 14)(8 35 73 15)(9 36 74 16)(10 25 75 17)(11 26 76 18)(12 27 77 19)(37 91 59 64)(38 92 60 65)(39 93 49 66)(40 94 50 67)(41 95 51 68)(42 96 52 69)(43 85 53 70)(44 86 54 71)(45 87 55 72)(46 88 56 61)(47 89 57 62)(48 90 58 63)
(13 39 93)(14 40 94)(15 41 95)(16 42 96)(17 43 85)(18 44 86)(19 45 87)(20 46 88)(21 47 89)(22 48 90)(23 37 91)(24 38 92)(25 53 70)(26 54 71)(27 55 72)(28 56 61)(29 57 62)(30 58 63)(31 59 64)(32 60 65)(33 49 66)(34 50 67)(35 51 68)(36 52 69)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6X 12A ··· 12H 12I ··· 12AF 12AG ··· 12AN order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 4 ··· 4 1 1 1 1 6 6 6 6 1 ··· 1 4 ··· 4 1 ··· 1 4 ··· 4 6 ··· 6

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 type + + - + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 SL2(𝔽3) SL2(𝔽3) C4.A4 C3×SL2(𝔽3) C3×C4.A4 A4 C2×A4 C3×A4 C4×A4 C6×A4 C12×A4 kernel C12×SL2(𝔽3) C6×SL2(𝔽3) C4×SL2(𝔽3) Q8×C12 C3×SL2(𝔽3) C2×SL2(𝔽3) C6×Q8 SL2(𝔽3) C3×Q8 C12 C12 C6 C4 C2 C2×C12 C2×C6 C2×C4 C6 C22 C2 # reps 1 1 6 2 2 6 2 12 4 2 4 6 12 12 1 1 2 2 2 4

Matrix representation of C12×SL2(𝔽3) in GL3(𝔽13) generated by

 6 0 0 0 3 0 0 0 3
,
 1 0 0 0 0 1 0 12 0
,
 1 0 0 0 9 10 0 10 4
,
 3 0 0 0 0 1 0 10 4
G:=sub<GL(3,GF(13))| [6,0,0,0,3,0,0,0,3],[1,0,0,0,0,12,0,1,0],[1,0,0,0,9,10,0,10,4],[3,0,0,0,0,10,0,1,4] >;

C12×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{12}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C12xSL(2,3)");
// GroupNames label

G:=SmallGroup(288,633);
// by ID

G=gap.SmallGroup(288,633);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,2,-2,126,1271,172,2280,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

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