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G = S3×C3⋊C16order 288 = 25·32

Direct product of S3 and C3⋊C16

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

Aliases: S3×C3⋊C16, C24.59D6, (C3×S3)⋊C16, C8.20S32, C33(S3×C16), (S3×C6).1C8, (S3×C8).3S3, C6.17(S3×C8), D6.2(C3⋊C8), C324(C2×C16), C3⋊C8.3Dic3, (S3×C24).3C2, (S3×C12).5C4, C12.94(C4×S3), C24.S36C2, Dic3.2(C3⋊C8), (C4×S3).4Dic3, (C3×Dic3).2C8, C4.16(S3×Dic3), (C3×C24).41C22, C12.27(C2×Dic3), C31(C2×C3⋊C16), C6.1(C2×C3⋊C8), C2.1(S3×C3⋊C8), (C3×C3⋊C16)⋊7C2, (C3×C3⋊C8).3C4, (C3×C6).12(C2×C8), (C3×C12).77(C2×C4), SmallGroup(288,189)

Series: Derived Chief Lower central Upper central

C1C32 — S3×C3⋊C16
C1C3C32C3×C6C3×C12C3×C24S3×C24 — S3×C3⋊C16
C32 — S3×C3⋊C16
C1C8

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 [×2], C3 [×2], C3, C4, C4, C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16 [×2], C2×C8, C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, C2×C16, C3×Dic3, C3×C12, S3×C6, C3⋊C16, C3⋊C16 [×3], 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 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, Dic3 [×2], D6 [×2], C16 [×2], C2×C8, C3⋊C8 [×2], C4×S3, C2×Dic3, C2×C16, S32, C3⋊C16 [×2], 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 26 43)(2 27 44)(3 28 45)(4 29 46)(5 30 47)(6 31 48)(7 32 33)(8 17 34)(9 18 35)(10 19 36)(11 20 37)(12 21 38)(13 22 39)(14 23 40)(15 24 41)(16 25 42)(49 66 86)(50 67 87)(51 68 88)(52 69 89)(53 70 90)(54 71 91)(55 72 92)(56 73 93)(57 74 94)(58 75 95)(59 76 96)(60 77 81)(61 78 82)(62 79 83)(63 80 84)(64 65 85)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 65)(16 66)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 93)(34 94)(35 95)(36 96)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)
(1 43 26)(2 27 44)(3 45 28)(4 29 46)(5 47 30)(6 31 48)(7 33 32)(8 17 34)(9 35 18)(10 19 36)(11 37 20)(12 21 38)(13 39 22)(14 23 40)(15 41 24)(16 25 42)(49 86 66)(50 67 87)(51 88 68)(52 69 89)(53 90 70)(54 71 91)(55 92 72)(56 73 93)(57 94 74)(58 75 95)(59 96 76)(60 77 81)(61 82 78)(62 79 83)(63 84 80)(64 65 85)
(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)| (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,33)(8,17,34)(9,18,35)(10,19,36)(11,20,37)(12,21,38)(13,22,39)(14,23,40)(15,24,41)(16,25,42)(49,66,86)(50,67,87)(51,68,88)(52,69,89)(53,70,90)(54,71,91)(55,72,92)(56,73,93)(57,74,94)(58,75,95)(59,76,96)(60,77,81)(61,78,82)(62,79,83)(63,80,84)(64,65,85), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,65)(16,66)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,93)(34,94)(35,95)(36,96)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92), (1,43,26)(2,27,44)(3,45,28)(4,29,46)(5,47,30)(6,31,48)(7,33,32)(8,17,34)(9,35,18)(10,19,36)(11,37,20)(12,21,38)(13,39,22)(14,23,40)(15,41,24)(16,25,42)(49,86,66)(50,67,87)(51,88,68)(52,69,89)(53,90,70)(54,71,91)(55,92,72)(56,73,93)(57,94,74)(58,75,95)(59,96,76)(60,77,81)(61,82,78)(62,79,83)(63,84,80)(64,65,85), (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( (1,26,43)(2,27,44)(3,28,45)(4,29,46)(5,30,47)(6,31,48)(7,32,33)(8,17,34)(9,18,35)(10,19,36)(11,20,37)(12,21,38)(13,22,39)(14,23,40)(15,24,41)(16,25,42)(49,66,86)(50,67,87)(51,68,88)(52,69,89)(53,70,90)(54,71,91)(55,72,92)(56,73,93)(57,74,94)(58,75,95)(59,76,96)(60,77,81)(61,78,82)(62,79,83)(63,80,84)(64,65,85), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,65)(16,66)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,93)(34,94)(35,95)(36,96)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92), (1,43,26)(2,27,44)(3,45,28)(4,29,46)(5,47,30)(6,31,48)(7,33,32)(8,17,34)(9,35,18)(10,19,36)(11,37,20)(12,21,38)(13,39,22)(14,23,40)(15,41,24)(16,25,42)(49,86,66)(50,67,87)(51,88,68)(52,69,89)(53,90,70)(54,71,91)(55,92,72)(56,73,93)(57,94,74)(58,75,95)(59,96,76)(60,77,81)(61,82,78)(62,79,83)(63,84,80)(64,65,85), (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([(1,26,43),(2,27,44),(3,28,45),(4,29,46),(5,30,47),(6,31,48),(7,32,33),(8,17,34),(9,18,35),(10,19,36),(11,20,37),(12,21,38),(13,22,39),(14,23,40),(15,24,41),(16,25,42),(49,66,86),(50,67,87),(51,68,88),(52,69,89),(53,70,90),(54,71,91),(55,72,92),(56,73,93),(57,74,94),(58,75,95),(59,76,96),(60,77,81),(61,78,82),(62,79,83),(63,80,84),(64,65,85)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,65),(16,66),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,93),(34,94),(35,95),(36,96),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,91),(48,92)], [(1,43,26),(2,27,44),(3,45,28),(4,29,46),(5,47,30),(6,31,48),(7,33,32),(8,17,34),(9,35,18),(10,19,36),(11,37,20),(12,21,38),(13,39,22),(14,23,40),(15,41,24),(16,25,42),(49,86,66),(50,67,87),(51,88,68),(52,69,89),(53,90,70),(54,71,91),(55,92,72),(56,73,93),(57,94,74),(58,75,95),(59,96,76),(60,77,81),(61,82,78),(62,79,83),(63,84,80),(64,65,85)], [(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 2A2B2C3A3B3C4A4B4C4D6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H16A···16H16I···16P24A···24H24I24J24K24L24M24N24O24P48A···48H
order122233344446666688888888121212121212121216···1616···1624···24242424242424242448···48
size113322411332246611113333222244663···39···92···2444466666···6

72 irreducible representations

dim111111111222222222224444
type++++++-+-+-
imageC1C2C2C2C4C4C8C8C16S3S3Dic3D6Dic3C3⋊C8C4×S3C3⋊C8C3⋊C16S3×C8S3×C16S32S3×Dic3S3×C3⋊C8S3×C3⋊C16
kernelS3×C3⋊C16C3×C3⋊C16C24.S3S3×C24C3×C3⋊C8S3×C12C3×Dic3S3×C6C3×S3C3⋊C16S3×C8C3⋊C8C24C4×S3Dic3C12D6S3C6C3C8C4C2C1
# reps1111224416111212228481124

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

0100
969600
0010
0001
,
1000
969600
00960
00096
,
1000
0100
0001
009696
,
1000
0100
00089
00890
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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