direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C3⋊C16, C24.59D6, (C3×S3)⋊C16, C8.20S32, C3⋊3(S3×C16), (S3×C6).1C8, (S3×C8).3S3, C6.17(S3×C8), D6.2(C3⋊C8), C32⋊4(C2×C16), C3⋊C8.3Dic3, (S3×C24).3C2, (S3×C12).5C4, C12.94(C4×S3), C24.S3⋊6C2, Dic3.2(C3⋊C8), (C4×S3).4Dic3, (C3×Dic3).2C8, C4.16(S3×Dic3), (C3×C24).41C22, C12.27(C2×Dic3), C3⋊1(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
| C32 — S3×C3⋊C16 |
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, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C2×C16, C3×Dic3, C3×C12, S3×C6, C3⋊C16, C3⋊C16, 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, C4, C22, S3, C8, C2×C4, Dic3, D6, C16, C2×C8, C3⋊C8, C4×S3, C2×Dic3, C2×C16, S32, C3⋊C16, S3×C8, C2×C3⋊C8, S3×Dic3, S3×C16, C2×C3⋊C16, S3×C3⋊C8, S3×C3⋊C16
►On 96 points
Generators in S96
(1 46 20)(2 47 21)(3 48 22)(4 33 23)(5 34 24)(6 35 25)(7 36 26)(8 37 27)(9 38 28)(10 39 29)(11 40 30)(12 41 31)(13 42 32)(14 43 17)(15 44 18)(16 45 19)(49 84 73)(50 85 74)(51 86 75)(52 87 76)(53 88 77)(54 89 78)(55 90 79)(56 91 80)(57 92 65)(58 93 66)(59 94 67)(60 95 68)(61 96 69)(62 81 70)(63 82 71)(64 83 72)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 81)(16 82)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 65)(30 66)(31 67)(32 68)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 49)(48 50)
(1 20 46)(2 47 21)(3 22 48)(4 33 23)(5 24 34)(6 35 25)(7 26 36)(8 37 27)(9 28 38)(10 39 29)(11 30 40)(12 41 31)(13 32 42)(14 43 17)(15 18 44)(16 45 19)(49 73 84)(50 85 74)(51 75 86)(52 87 76)(53 77 88)(54 89 78)(55 79 90)(56 91 80)(57 65 92)(58 93 66)(59 67 94)(60 95 68)(61 69 96)(62 81 70)(63 71 82)(64 83 72)
(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)| (1,46,20)(2,47,21)(3,48,22)(4,33,23)(5,34,24)(6,35,25)(7,36,26)(8,37,27)(9,38,28)(10,39,29)(11,40,30)(12,41,31)(13,42,32)(14,43,17)(15,44,18)(16,45,19)(49,84,73)(50,85,74)(51,86,75)(52,87,76)(53,88,77)(54,89,78)(55,90,79)(56,91,80)(57,92,65)(58,93,66)(59,94,67)(60,95,68)(61,96,69)(62,81,70)(63,82,71)(64,83,72), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,81)(16,82)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,65)(30,66)(31,67)(32,68)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,49)(48,50), (1,20,46)(2,47,21)(3,22,48)(4,33,23)(5,24,34)(6,35,25)(7,26,36)(8,37,27)(9,28,38)(10,39,29)(11,30,40)(12,41,31)(13,32,42)(14,43,17)(15,18,44)(16,45,19)(49,73,84)(50,85,74)(51,75,86)(52,87,76)(53,77,88)(54,89,78)(55,79,90)(56,91,80)(57,65,92)(58,93,66)(59,67,94)(60,95,68)(61,69,96)(62,81,70)(63,71,82)(64,83,72), (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( (1,46,20)(2,47,21)(3,48,22)(4,33,23)(5,34,24)(6,35,25)(7,36,26)(8,37,27)(9,38,28)(10,39,29)(11,40,30)(12,41,31)(13,42,32)(14,43,17)(15,44,18)(16,45,19)(49,84,73)(50,85,74)(51,86,75)(52,87,76)(53,88,77)(54,89,78)(55,90,79)(56,91,80)(57,92,65)(58,93,66)(59,94,67)(60,95,68)(61,96,69)(62,81,70)(63,82,71)(64,83,72), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,81)(16,82)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,65)(30,66)(31,67)(32,68)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,49)(48,50), (1,20,46)(2,47,21)(3,22,48)(4,33,23)(5,24,34)(6,35,25)(7,26,36)(8,37,27)(9,28,38)(10,39,29)(11,30,40)(12,41,31)(13,32,42)(14,43,17)(15,18,44)(16,45,19)(49,73,84)(50,85,74)(51,75,86)(52,87,76)(53,77,88)(54,89,78)(55,79,90)(56,91,80)(57,65,92)(58,93,66)(59,67,94)(60,95,68)(61,69,96)(62,81,70)(63,71,82)(64,83,72), (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([[(1,46,20),(2,47,21),(3,48,22),(4,33,23),(5,34,24),(6,35,25),(7,36,26),(8,37,27),(9,38,28),(10,39,29),(11,40,30),(12,41,31),(13,42,32),(14,43,17),(15,44,18),(16,45,19),(49,84,73),(50,85,74),(51,86,75),(52,87,76),(53,88,77),(54,89,78),(55,90,79),(56,91,80),(57,92,65),(58,93,66),(59,94,67),(60,95,68),(61,96,69),(62,81,70),(63,82,71),(64,83,72)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,81),(16,82),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,65),(30,66),(31,67),(32,68),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,49),(48,50)], [(1,20,46),(2,47,21),(3,22,48),(4,33,23),(5,24,34),(6,35,25),(7,26,36),(8,37,27),(9,28,38),(10,39,29),(11,30,40),(12,41,31),(13,32,42),(14,43,17),(15,18,44),(16,45,19),(49,73,84),(50,85,74),(51,75,86),(52,87,76),(53,77,88),(54,89,78),(55,79,90),(56,91,80),(57,65,92),(58,93,66),(59,67,94),(60,95,68),(61,69,96),(62,81,70),(63,71,82),(64,83,72)], [(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 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 16I | ··· | 16P | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 24M | 24N | 24O | 24P | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 3 | ··· | 3 | 9 | ··· | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C4×S3 | C3⋊C8 | C3⋊C16 | S3×C8 | S3×C16 | S32 | S3×Dic3 | S3×C3⋊C8 | S3×C3⋊C16 |
| kernel | S3×C3⋊C16 | C3×C3⋊C16 | C24.S3 | S3×C24 | C3×C3⋊C8 | S3×C12 | C3×Dic3 | S3×C6 | C3×S3 | C3⋊C16 | S3×C8 | C3⋊C8 | C24 | C4×S3 | Dic3 | C12 | D6 | S3 | C6 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 8 | 4 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of S3×C3⋊C16 ►in GL4(𝔽97) generated by
| 0 | 1 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 96 | 96 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 89 |
| 0 | 0 | 89 | 0 |
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