metabelian, supersoluble, monomial
Aliases: D24⋊3S3, C32⋊2D16, C24.14D6, C8.13S32, (C3×C6).6D8, (C3×D24)⋊5C2, C3⋊2(C3⋊D16), C6.8(D4⋊S3), (C3×C12).21D4, C24.S3⋊1C2, (C3×C24).6C22, C4.1(D6⋊S3), C12.21(C3⋊D4), C2.3(C32⋊2D8), SmallGroup(288,193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊2D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 302 in 61 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, C12, D6, C2×C6, C16, D8, C3×S3, C3×C6, C24, C24, D12, C3×D4, D16, C3×C12, S3×C6, C3⋊C16, D24, C3×D8, C3×C24, C3×D12, C3⋊D16, C24.S3, C3×D24, C32⋊2D16
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊D4, D16, S32, D4⋊S3, D6⋊S3, C3⋊D16, C32⋊2D8, C32⋊2D16
►On 96 points
Generators in S96
(1 95 27)(2 28 96)(3 81 29)(4 30 82)(5 83 31)(6 32 84)(7 85 17)(8 18 86)(9 87 19)(10 20 88)(11 89 21)(12 22 90)(13 91 23)(14 24 92)(15 93 25)(16 26 94)(33 51 72)(34 73 52)(35 53 74)(36 75 54)(37 55 76)(38 77 56)(39 57 78)(40 79 58)(41 59 80)(42 65 60)(43 61 66)(44 67 62)(45 63 68)(46 69 64)(47 49 70)(48 71 50)
(1 27 95)(2 96 28)(3 29 81)(4 82 30)(5 31 83)(6 84 32)(7 17 85)(8 86 18)(9 19 87)(10 88 20)(11 21 89)(12 90 22)(13 23 91)(14 92 24)(15 25 93)(16 94 26)(33 51 72)(34 73 52)(35 53 74)(36 75 54)(37 55 76)(38 77 56)(39 57 78)(40 79 58)(41 59 80)(42 65 60)(43 61 66)(44 67 62)(45 63 68)(46 69 64)(47 49 70)(48 71 50)
(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 66)(2 65)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 64)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 56)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)
G:=sub<Sym(96)| (1,95,27)(2,28,96)(3,81,29)(4,30,82)(5,83,31)(6,32,84)(7,85,17)(8,18,86)(9,87,19)(10,20,88)(11,89,21)(12,22,90)(13,91,23)(14,24,92)(15,93,25)(16,26,94)(33,51,72)(34,73,52)(35,53,74)(36,75,54)(37,55,76)(38,77,56)(39,57,78)(40,79,58)(41,59,80)(42,65,60)(43,61,66)(44,67,62)(45,63,68)(46,69,64)(47,49,70)(48,71,50), (1,27,95)(2,96,28)(3,29,81)(4,82,30)(5,31,83)(6,84,32)(7,17,85)(8,86,18)(9,19,87)(10,88,20)(11,21,89)(12,90,22)(13,23,91)(14,92,24)(15,25,93)(16,94,26)(33,51,72)(34,73,52)(35,53,74)(36,75,54)(37,55,76)(38,77,56)(39,57,78)(40,79,58)(41,59,80)(42,65,60)(43,61,66)(44,67,62)(45,63,68)(46,69,64)(47,49,70)(48,71,50), (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), (1,66)(2,65)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)(33,89)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90)>;
G:=Group( (1,95,27)(2,28,96)(3,81,29)(4,30,82)(5,83,31)(6,32,84)(7,85,17)(8,18,86)(9,87,19)(10,20,88)(11,89,21)(12,22,90)(13,91,23)(14,24,92)(15,93,25)(16,26,94)(33,51,72)(34,73,52)(35,53,74)(36,75,54)(37,55,76)(38,77,56)(39,57,78)(40,79,58)(41,59,80)(42,65,60)(43,61,66)(44,67,62)(45,63,68)(46,69,64)(47,49,70)(48,71,50), (1,27,95)(2,96,28)(3,29,81)(4,82,30)(5,31,83)(6,84,32)(7,17,85)(8,86,18)(9,19,87)(10,88,20)(11,21,89)(12,90,22)(13,23,91)(14,92,24)(15,25,93)(16,94,26)(33,51,72)(34,73,52)(35,53,74)(36,75,54)(37,55,76)(38,77,56)(39,57,78)(40,79,58)(41,59,80)(42,65,60)(43,61,66)(44,67,62)(45,63,68)(46,69,64)(47,49,70)(48,71,50), (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), (1,66)(2,65)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)(33,89)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90) );
G=PermutationGroup([[(1,95,27),(2,28,96),(3,81,29),(4,30,82),(5,83,31),(6,32,84),(7,85,17),(8,18,86),(9,87,19),(10,20,88),(11,89,21),(12,22,90),(13,91,23),(14,24,92),(15,93,25),(16,26,94),(33,51,72),(34,73,52),(35,53,74),(36,75,54),(37,55,76),(38,77,56),(39,57,78),(40,79,58),(41,59,80),(42,65,60),(43,61,66),(44,67,62),(45,63,68),(46,69,64),(47,49,70),(48,71,50)], [(1,27,95),(2,96,28),(3,29,81),(4,82,30),(5,31,83),(6,84,32),(7,17,85),(8,86,18),(9,19,87),(10,88,20),(11,21,89),(12,90,22),(13,23,91),(14,92,24),(15,25,93),(16,94,26),(33,51,72),(34,73,52),(35,53,74),(36,75,54),(37,55,76),(38,77,56),(39,57,78),(40,79,58),(41,59,80),(42,65,60),(43,61,66),(44,67,62),(45,63,68),(46,69,64),(47,49,70),(48,71,50)], [(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)], [(1,66),(2,65),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,64),(25,63),(26,62),(27,61),(28,60),(29,59),(30,58),(31,57),(32,56),(33,89),(34,88),(35,87),(36,86),(37,85),(38,84),(39,83),(40,82),(41,81),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 24 | 24 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 24 | 24 | 24 | 24 | 2 | 2 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | C3⋊D4 | D16 | S32 | D4⋊S3 | D6⋊S3 | C3⋊D16 | C32⋊2D8 | C32⋊2D16 |
| kernel | C32⋊2D16 | C24.S3 | C3×D24 | D24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 4 |
Matrix representation of C32⋊2D16 ►in GL6(𝔽97)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 |
| 0 | 0 | 0 | 0 | 1 | 96 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 95 | 0 | 0 | 0 | 0 |
| 2 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 79 | 68 | 0 | 0 | 0 | 0 |
| 68 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,2,0,0,0,0,95,26,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[79,68,0,0,0,0,68,18,0,0,0,0,0,0,0,96,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C32⋊2D16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2D_{16} % in TeX
G:=Group("C3^2:2D16"); // GroupNames label
G:=SmallGroup(288,193);
// by ID
G=gap.SmallGroup(288,193);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,254,135,142,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations