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G = C32⋊S3order 192 = 26·3

2nd semidirect product of C32 and S3 acting via S3/C3=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C962C2, C322S3, C31SD64, C8.6D12, C6.2D16, C4.2D24, C2.4D48, D48.1C2, C12.27D8, C24.56D4, C16.14D6, Dic241C2, C48.15C22, SmallGroup(192,8)

Series: Derived Chief Lower central Upper central

C1C48 — C32⋊S3
C1C3C6C12C24C48D48 — C32⋊S3
C3C6C12C24C48 — C32⋊S3
C1C2C4C8C16C32

Generators and relations for C32⋊S3
 G = < a,b,c | a32=b3=c2=1, ab=ba, cac=a15, cbc=b-1 >

48C2
24C22
24C4
16S3
12Q8
12D4
8D6
8Dic3
6Q16
6D8
4D12
4Dic6
3Q32
3D16
2Dic12
2D24
3SD64

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

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

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

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

51 conjugacy classes

class 1 2A2B 3 4A4B 6 8A8B12A12B16A16B16C16D24A24B24C24D32A···32H48A···48H96A···96P
order1223446881212161616162424242432···3248···4896···96
size1148224822222222222222···22···22···2

51 irreducible representations

dim11112222222222
type++++++++++++
imageC1C2C2C2S3D4D6D8D12D16D24SD64D48C32⋊S3
kernelC32⋊S3C96D48Dic24C32C24C16C12C8C6C4C3C2C1
# reps111111122448816

Matrix representation of C32⋊S3 in GL2(𝔽47) generated by

4026
2611
,
267
720
,
126
046
G:=sub<GL(2,GF(47))| [40,26,26,11],[26,7,7,20],[1,0,26,46] >;

C32⋊S3 in GAP, Magma, Sage, TeX

C_{32}\rtimes S_3
% in TeX

G:=Group("C32:S3");
// GroupNames label

G:=SmallGroup(192,8);
// by ID

G=gap.SmallGroup(192,8);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,926,142,1571,80,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^32=b^3=c^2=1,a*b=b*a,c*a*c=a^15,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C32⋊S3 in TeX

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