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## G = S3×C3⋊C16order 288 = 25·32

### Direct product of S3 and C3⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×C3⋊C16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — S3×C24 — S3×C3⋊C16
 Lower central C32 — S3×C3⋊C16
 Upper central C1 — C8

Generators and relations for S3×C3⋊C16
G = < a,b,c,d | a3=b2=c3=d16=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 126 in 61 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, 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, C3⋊C16, C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, S3×C16, C2×C3⋊C16, C3×C3⋊C16, C24.S3, S3×C24, S3×C3⋊C16
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C16, C2×C8, C3⋊C8, C4×S3, C2×Dic3, C2×C16, S32, C3⋊C16, S3×C8, C2×C3⋊C8, S3×Dic3, S3×C16, C2×C3⋊C16, S3×C3⋊C8, S3×C3⋊C16

Smallest permutation representation of S3×C3⋊C16
On 96 points
Generators in S96
(1 46 20)(2 47 21)(3 48 22)(4 33 23)(5 34 24)(6 35 25)(7 36 26)(8 37 27)(9 38 28)(10 39 29)(11 40 30)(12 41 31)(13 42 32)(14 43 17)(15 44 18)(16 45 19)(49 84 73)(50 85 74)(51 86 75)(52 87 76)(53 88 77)(54 89 78)(55 90 79)(56 91 80)(57 92 65)(58 93 66)(59 94 67)(60 95 68)(61 96 69)(62 81 70)(63 82 71)(64 83 72)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 81)(16 82)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 65)(30 66)(31 67)(32 68)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 49)(48 50)
(1 20 46)(2 47 21)(3 22 48)(4 33 23)(5 24 34)(6 35 25)(7 26 36)(8 37 27)(9 28 38)(10 39 29)(11 30 40)(12 41 31)(13 32 42)(14 43 17)(15 18 44)(16 45 19)(49 73 84)(50 85 74)(51 75 86)(52 87 76)(53 77 88)(54 89 78)(55 79 90)(56 91 80)(57 65 92)(58 93 66)(59 67 94)(60 95 68)(61 69 96)(62 81 70)(63 71 82)(64 83 72)
(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)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 12G 12H 16A ··· 16H 16I ··· 16P 24A ··· 24H 24I 24J 24K 24L 24M 24N 24O 24P 48A ··· 48H order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 16 ··· 16 16 ··· 16 24 ··· 24 24 24 24 24 24 24 24 24 48 ··· 48 size 1 1 3 3 2 2 4 1 1 3 3 2 2 4 6 6 1 1 1 1 3 3 3 3 2 2 2 2 4 4 6 6 3 ··· 3 9 ··· 9 2 ··· 2 4 4 4 4 6 6 6 6 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C8 C8 C16 S3 S3 Dic3 D6 Dic3 C3⋊C8 C4×S3 C3⋊C8 C3⋊C16 S3×C8 S3×C16 S32 S3×Dic3 S3×C3⋊C8 S3×C3⋊C16 kernel S3×C3⋊C16 C3×C3⋊C16 C24.S3 S3×C24 C3×C3⋊C8 S3×C12 C3×Dic3 S3×C6 C3×S3 C3⋊C16 S3×C8 C3⋊C8 C24 C4×S3 Dic3 C12 D6 S3 C6 C3 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 1 1 2 1 2 2 2 8 4 8 1 1 2 4

Matrix representation of S3×C3⋊C16 in GL4(𝔽97) generated by

 0 1 0 0 96 96 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 96 96 0 0 0 0 96 0 0 0 0 96
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 96 96
,
 1 0 0 0 0 1 0 0 0 0 0 89 0 0 89 0
G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,0,1,0,0,0,0,1],[1,96,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[1,0,0,0,0,1,0,0,0,0,0,96,0,0,1,96],[1,0,0,0,0,1,0,0,0,0,0,89,0,0,89,0] >;

S3×C3⋊C16 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes C_{16}
% in TeX

G:=Group("S3xC3:C16");
// GroupNames label

G:=SmallGroup(288,189);
// by ID

G=gap.SmallGroup(288,189);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,36,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^3=d^16=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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