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## G = S3×C48order 288 = 25·32

### Direct product of C48 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C48
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — S3×C24 — S3×C48
 Lower central C3 — S3×C48
 Upper central C1 — C48

Generators and relations for S3×C48
G = < a,b,c | a48=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 102 in 61 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C16, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C2×C16, C3×Dic3, C3×C12, S3×C6, C3⋊C16, C48, C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, S3×C16, C2×C48, C3×C3⋊C16, C3×C48, S3×C24, S3×C48
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C16, C2×C8, C3×S3, C24, C4×S3, C2×C12, C2×C16, S3×C6, C48, S3×C8, C2×C24, S3×C12, S3×C16, C2×C48, S3×C24, S3×C48

Smallest permutation representation of S3×C48
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 17 33)(2 18 34)(3 19 35)(4 20 36)(5 21 37)(6 22 38)(7 23 39)(8 24 40)(9 25 41)(10 26 42)(11 27 43)(12 28 44)(13 29 45)(14 30 46)(15 31 47)(16 32 48)(49 81 65)(50 82 66)(51 83 67)(52 84 68)(53 85 69)(54 86 70)(55 87 71)(56 88 72)(57 89 73)(58 90 74)(59 91 75)(60 92 76)(61 93 77)(62 94 78)(63 95 79)(64 96 80)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 49)(47 50)(48 51)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 16A ··· 16H 16I ··· 16P 24A ··· 24H 24I ··· 24T 24U ··· 24AB 48A ··· 48P 48Q ··· 48AN 48AO ··· 48BD order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 16 ··· 16 16 ··· 16 24 ··· 24 24 ··· 24 24 ··· 24 48 ··· 48 48 ··· 48 48 ··· 48 size 1 1 3 3 1 1 2 2 2 1 1 3 3 1 1 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 1 1 1 1 2 ··· 2 3 3 3 3 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C8 C12 C12 C16 C24 C24 C48 S3 D6 C3×S3 C4×S3 S3×C6 S3×C8 S3×C12 S3×C16 S3×C24 S3×C48 kernel S3×C48 C3×C3⋊C16 C3×C48 S3×C24 S3×C16 C3×C3⋊C8 S3×C12 C3⋊C16 C48 S3×C8 C3×Dic3 S3×C6 C3⋊C8 C4×S3 C3×S3 Dic3 D6 S3 C48 C24 C16 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 4 4 16 8 8 32 1 1 2 2 2 4 4 8 8 16

Matrix representation of S3×C48 in GL3(𝔽97) generated by

 66 0 0 0 88 0 0 0 88
,
 1 0 0 0 35 0 0 61 61
,
 96 0 0 0 96 60 0 0 1
G:=sub<GL(3,GF(97))| [66,0,0,0,88,0,0,0,88],[1,0,0,0,35,61,0,0,61],[96,0,0,0,96,0,0,60,1] >;

S3×C48 in GAP, Magma, Sage, TeX

S_3\times C_{48}
% in TeX

G:=Group("S3xC48");
// GroupNames label

G:=SmallGroup(288,231);
// by ID

G=gap.SmallGroup(288,231);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,92,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^48=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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