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G = S3xC2xC16order 192 = 26·3

Direct product of C2xC16 and S3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3xC2xC16, C48:12C22, C24.71C23, C6:1(C2xC16), (C2xC48):15C2, (C4xS3).5C8, (S3xC8).6C4, C4.23(S3xC8), C8.43(C4xS3), C3:1(C22xC16), D6.9(C2xC8), C3:C16:13C22, C12.28(C2xC8), C24.64(C2xC4), (C2xC8).341D6, (C22xS3).5C8, C22.13(S3xC8), C6.12(C22xC8), C8.57(C22xS3), (C2xDic3).8C8, (S3xC8).20C22, Dic3.10(C2xC8), (C2xC24).427C22, C12.128(C22xC4), C2.2(S3xC2xC8), (C2xC3:C16):17C2, (C2xC3:C8).18C4, C3:C8.25(C2xC4), (S3xC2xC4).24C4, (S3xC2xC8).20C2, C4.102(S3xC2xC4), (C2xC6).14(C2xC8), (C4xS3).38(C2xC4), (C2xC4).175(C4xS3), (C2xC12).248(C2xC4), SmallGroup(192,458)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC2xC16
C1C3C6C12C24S3xC8S3xC2xC8 — S3xC2xC16
C3 — S3xC2xC16
C1C2xC16

Generators and relations for S3xC2xC16
 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, C2xC4, C2xC4, C23, Dic3, C12, D6, C2xC6, C16, C16, C2xC8, C2xC8, C22xC4, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C22xS3, C2xC16, C2xC16, C22xC8, C3:C16, C48, S3xC8, C2xC3:C8, C2xC24, S3xC2xC4, C22xC16, S3xC16, C2xC3:C16, C2xC48, S3xC2xC8, S3xC2xC16
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, C23, D6, C16, C2xC8, C22xC4, C4xS3, C22xS3, C2xC16, C22xC8, S3xC8, S3xC2xC4, C22xC16, S3xC16, S3xC2xC8, S3xC2xC16

Smallest permutation representation of S3xC2xC16
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 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H6A6B6C8A···8H8I···8P12A12B12C12D16A···16P16Q···16AF24A···24H48A···48P
order122222223444444446668···88···81212121216···1616···1624···2448···48
size111133332111133332221···13···322221···13···32···22···2

96 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C4C4C4C8C8C8C16S3D6D6C4xS3C4xS3S3xC8S3xC8S3xC16
kernelS3xC2xC16S3xC16C2xC3:C16C2xC48S3xC2xC8S3xC8C2xC3:C8S3xC2xC4C4xS3C2xDic3C22xS3D6C2xC16C16C2xC8C8C2xC4C4C22C2
# reps1411142284432121224416

Matrix representation of S3xC2xC16 in GL3(F97) generated by

100
0960
0096
,
1200
0470
0047
,
100
09696
010
,
100
010
09696
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] >;

S3xC2xC16 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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