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## G = S3×C2×C16order 192 = 26·3

### Direct product of C2×C16 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C16
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×C2×C8 — S3×C2×C16
 Lower central C3 — S3×C2×C16
 Upper central C1 — C2×C16

Generators and relations for S3×C2×C16
G = < a,b,c,d | a2=b16=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 168 in 98 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C16, C16, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C16, C2×C16, C22×C8, C3⋊C16, C48, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C22×C16, S3×C16, C2×C3⋊C16, C2×C48, S3×C2×C8, S3×C2×C16
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C16, C2×C8, C22×C4, C4×S3, C22×S3, C2×C16, C22×C8, S3×C8, S3×C2×C4, C22×C16, S3×C16, S3×C2×C8, S3×C2×C16

Smallest permutation representation of S3×C2×C16
On 96 points
Generators in S96
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 65)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 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 86 80)(2 87 65)(3 88 66)(4 89 67)(5 90 68)(6 91 69)(7 92 70)(8 93 71)(9 94 72)(10 95 73)(11 96 74)(12 81 75)(13 82 76)(14 83 77)(15 84 78)(16 85 79)(17 60 40)(18 61 41)(19 62 42)(20 63 43)(21 64 44)(22 49 45)(23 50 46)(24 51 47)(25 52 48)(26 53 33)(27 54 34)(28 55 35)(29 56 36)(30 57 37)(31 58 38)(32 59 39)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 48)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 95)(66 96)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A ··· 8H 8I ··· 8P 12A 12B 12C 12D 16A ··· 16P 16Q ··· 16AF 24A ··· 24H 48A ··· 48P order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 8 ··· 8 8 ··· 8 12 12 12 12 16 ··· 16 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 3 3 3 3 2 1 1 1 1 3 3 3 3 2 2 2 1 ··· 1 3 ··· 3 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 C16 S3 D6 D6 C4×S3 C4×S3 S3×C8 S3×C8 S3×C16 kernel S3×C2×C16 S3×C16 C2×C3⋊C16 C2×C48 S3×C2×C8 S3×C8 C2×C3⋊C8 S3×C2×C4 C4×S3 C2×Dic3 C22×S3 D6 C2×C16 C16 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 8 4 4 32 1 2 1 2 2 4 4 16

Matrix representation of S3×C2×C16 in GL3(𝔽97) generated by

 1 0 0 0 96 0 0 0 96
,
 12 0 0 0 47 0 0 0 47
,
 1 0 0 0 96 96 0 1 0
,
 1 0 0 0 1 0 0 96 96
G:=sub<GL(3,GF(97))| [1,0,0,0,96,0,0,0,96],[12,0,0,0,47,0,0,0,47],[1,0,0,0,96,1,0,96,0],[1,0,0,0,1,96,0,0,96] >;

S3×C2×C16 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{16}
% in TeX

G:=Group("S3xC2xC16");
// GroupNames label

G:=SmallGroup(192,458);
// by ID

G=gap.SmallGroup(192,458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,58,80,102,6278]);
// Polycyclic

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

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