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G = S3×C32order 192 = 26·3

Direct product of C32 and S3

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

Aliases: S3×C32, C965C2, D6.2C16, C16.19D6, C48.24C22, Dic3.2C16, C3⋊C326C2, C31(C2×C32), C3⋊C8.4C8, C3⋊C16.3C4, (C4×S3).4C8, (S3×C8).5C4, C4.16(S3×C8), C8.36(C4×S3), C2.1(S3×C16), C6.1(C2×C16), (S3×C16).3C2, C24.57(C2×C4), C12.21(C2×C8), SmallGroup(192,5)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C32
C1C3C6C12C24C48S3×C16 — S3×C32
C3 — S3×C32
C1C32

Generators and relations for S3×C32
 G = < a,b,c | a32=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C4
3C2×C4
3C8
3C2×C8
3C16
3C2×C16
3C32
3C2×C32

Smallest permutation representation of S3×C32
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 93 35)(2 94 36)(3 95 37)(4 96 38)(5 65 39)(6 66 40)(7 67 41)(8 68 42)(9 69 43)(10 70 44)(11 71 45)(12 72 46)(13 73 47)(14 74 48)(15 75 49)(16 76 50)(17 77 51)(18 78 52)(19 79 53)(20 80 54)(21 81 55)(22 82 56)(23 83 57)(24 84 58)(25 85 59)(26 86 60)(27 87 61)(28 88 62)(29 89 63)(30 90 64)(31 91 33)(32 92 34)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)

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

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

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

96 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 8A8B8C8D8E8F8G8H12A12B16A···16H16I···16P24A24B24C24D32A···32P32Q···32AF48A···48H96A···96P
order122234444688888888121216···1616···162424242432···3232···3248···4896···96
size113321133211113333221···13···322221···13···32···22···2

96 irreducible representations

dim11111111111222222
type++++++
imageC1C2C2C2C4C4C8C8C16C16C32S3D6C4×S3S3×C8S3×C16S3×C32
kernelS3×C32C3⋊C32C96S3×C16C3⋊C16S3×C8C3⋊C8C4×S3Dic3D6S3C32C16C8C4C2C1
# reps1111224488321124816

Matrix representation of S3×C32 in GL2(𝔽97) generated by

460
046
,
096
196
,
096
960
G:=sub<GL(2,GF(97))| [46,0,0,46],[0,1,96,96],[0,96,96,0] >;

S3×C32 in GAP, Magma, Sage, TeX

S_3\times C_{32}
% in TeX

G:=Group("S3xC32");
// GroupNames label

G:=SmallGroup(192,5);
// by ID

G=gap.SmallGroup(192,5);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,36,58,80,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^32=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C32 in TeX

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