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## G = C3×C42.S3order 288 = 25·32

### Direct product of C3 and C42.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C42.S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×C3⋊C8 — C3×C42.S3
 Lower central C3 — C6 — C3×C42.S3
 Upper central C1 — C2×C12 — C4×C12

Generators and relations for C3×C42.S3
G = < a,b,c,d | a3=b6=d4=1, c4=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 146 in 95 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, C32, C12, C12, C2×C6, C2×C6, C42, C2×C8, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C8⋊C4, C3×C12, C3×C12, C62, C2×C3⋊C8, C4×C12, C4×C12, C2×C24, C3×C3⋊C8, C6×C12, C6×C12, C42.S3, C3×C8⋊C4, C6×C3⋊C8, C122, C3×C42.S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, M4(2), C3×S3, C4×S3, C2×Dic3, C2×C12, C8⋊C4, C3×Dic3, S3×C6, C4.Dic3, C4×Dic3, C4×C12, C3×M4(2), S3×C12, C6×Dic3, C42.S3, C3×C8⋊C4, C3×C4.Dic3, Dic3×C12, C3×C42.S3

Smallest permutation representation of C3×C42.S3
On 96 points
Generators in S96
(1 73 67)(2 74 68)(3 75 69)(4 76 70)(5 77 71)(6 78 72)(7 79 65)(8 80 66)(9 25 33)(10 26 34)(11 27 35)(12 28 36)(13 29 37)(14 30 38)(15 31 39)(16 32 40)(17 88 90)(18 81 91)(19 82 92)(20 83 93)(21 84 94)(22 85 95)(23 86 96)(24 87 89)(41 55 57)(42 56 58)(43 49 59)(44 50 60)(45 51 61)(46 52 62)(47 53 63)(48 54 64)
(1 71 73 5 67 77)(2 78 68 6 74 72)(3 65 75 7 69 79)(4 80 70 8 76 66)(9 37 25 13 33 29)(10 30 34 14 26 38)(11 39 27 15 35 31)(12 32 36 16 28 40)(17 84 90 21 88 94)(18 95 81 22 91 85)(19 86 92 23 82 96)(20 89 83 24 93 87)(41 61 55 45 57 51)(42 52 58 46 56 62)(43 63 49 47 59 53)(44 54 60 48 50 64)
(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 47 22 15)(2 44 23 12)(3 41 24 9)(4 46 17 14)(5 43 18 11)(6 48 19 16)(7 45 20 13)(8 42 21 10)(25 75 55 87)(26 80 56 84)(27 77 49 81)(28 74 50 86)(29 79 51 83)(30 76 52 88)(31 73 53 85)(32 78 54 82)(33 69 57 89)(34 66 58 94)(35 71 59 91)(36 68 60 96)(37 65 61 93)(38 70 62 90)(39 67 63 95)(40 72 64 92)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 8A ··· 8H 12A ··· 12H 12I ··· 12AZ 24A ··· 24P order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 M4(2) C3×S3 C4×S3 C3×Dic3 S3×C6 C4.Dic3 C3×M4(2) S3×C12 C3×C4.Dic3 kernel C3×C42.S3 C6×C3⋊C8 C122 C42.S3 C3×C3⋊C8 C6×C12 C2×C3⋊C8 C4×C12 C3⋊C8 C2×C12 C4×C12 C2×C12 C2×C12 C3×C6 C42 C12 C2×C4 C2×C4 C6 C6 C4 C2 # reps 1 2 1 2 8 4 4 2 16 8 1 2 1 4 2 4 4 2 8 8 8 16

Matrix representation of C3×C42.S3 in GL3(𝔽73) generated by

 8 0 0 0 64 0 0 0 64
,
 1 0 0 0 65 55 0 0 9
,
 1 0 0 0 54 18 0 22 19
,
 27 0 0 0 1 15 0 0 72
G:=sub<GL(3,GF(73))| [8,0,0,0,64,0,0,0,64],[1,0,0,0,65,0,0,55,9],[1,0,0,0,54,22,0,18,19],[27,0,0,0,1,0,0,15,72] >;

C3×C42.S3 in GAP, Magma, Sage, TeX

C_3\times C_4^2.S_3
% in TeX

G:=Group("C3xC4^2.S3");
// GroupNames label

G:=SmallGroup(288,237);
// by ID

G=gap.SmallGroup(288,237);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,176,136,9414]);
// Polycyclic

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

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