direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C3⋊D16, D8⋊7D6, C6⋊2D16, C24.20D4, C12.20D8, D24⋊13C22, C24.23C23, C3⋊3(C2×D16), (C6×D8)⋊4C2, (C2×D8)⋊1S3, C3⋊C16⋊7C22, C6.62(C2×D8), (C2×C6).41D8, (C2×D24)⋊17C2, C4.8(D4⋊S3), (C2×C8).233D6, (C3×D8)⋊7C22, (C2×C12).179D4, C12.159(C2×D4), C8.13(C3⋊D4), C8.29(C22×S3), (C2×C24).85C22, C22.21(D4⋊S3), (C2×C3⋊C16)⋊6C2, C4.1(C2×C3⋊D4), C2.17(C2×D4⋊S3), (C2×C4).142(C3⋊D4), SmallGroup(192,705)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3⋊D16
G = < a,b,c,d | a2=b3=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 408 in 98 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, C23, C12, D6, C2×C6, C2×C6, C16, C2×C8, D8, D8, C2×D4, C24, D12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C16, D16, C2×D8, C2×D8, C3⋊C16, D24, D24, C2×C24, C3×D8, C3×D8, C2×D12, C6×D4, C2×D16, C2×C3⋊C16, C3⋊D16, C2×D24, C6×D8, C2×C3⋊D16
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, D16, C2×D8, D4⋊S3, C2×C3⋊D4, C2×D16, C3⋊D16, C2×D4⋊S3, C2×C3⋊D16
►On 96 points
Generators in S96
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 49)(13 50)(14 51)(15 52)(16 53)(17 79)(18 80)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)
(1 41 71)(2 72 42)(3 43 73)(4 74 44)(5 45 75)(6 76 46)(7 47 77)(8 78 48)(9 33 79)(10 80 34)(11 35 65)(12 66 36)(13 37 67)(14 68 38)(15 39 69)(16 70 40)(17 62 90)(18 91 63)(19 64 92)(20 93 49)(21 50 94)(22 95 51)(23 52 96)(24 81 53)(25 54 82)(26 83 55)(27 56 84)(28 85 57)(29 58 86)(30 87 59)(31 60 88)(32 89 61)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 80)(49 59)(50 58)(51 57)(52 56)(53 55)(60 64)(61 63)
G:=sub<Sym(96)| (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,49)(13,50)(14,51)(15,52)(16,53)(17,79)(18,80)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89), (1,41,71)(2,72,42)(3,43,73)(4,74,44)(5,45,75)(6,76,46)(7,47,77)(8,78,48)(9,33,79)(10,80,34)(11,35,65)(12,66,36)(13,37,67)(14,68,38)(15,39,69)(16,70,40)(17,62,90)(18,91,63)(19,64,92)(20,93,49)(21,50,94)(22,95,51)(23,52,96)(24,81,53)(25,54,82)(26,83,55)(27,56,84)(28,85,57)(29,58,86)(30,87,59)(31,60,88)(32,89,61), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,80)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63)>;
G:=Group( (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,49)(13,50)(14,51)(15,52)(16,53)(17,79)(18,80)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89), (1,41,71)(2,72,42)(3,43,73)(4,74,44)(5,45,75)(6,76,46)(7,47,77)(8,78,48)(9,33,79)(10,80,34)(11,35,65)(12,66,36)(13,37,67)(14,68,38)(15,39,69)(16,70,40)(17,62,90)(18,91,63)(19,64,92)(20,93,49)(21,50,94)(22,95,51)(23,52,96)(24,81,53)(25,54,82)(26,83,55)(27,56,84)(28,85,57)(29,58,86)(30,87,59)(31,60,88)(32,89,61), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,80)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63) );
G=PermutationGroup([[(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,49),(13,50),(14,51),(15,52),(16,53),(17,79),(18,80),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89)], [(1,41,71),(2,72,42),(3,43,73),(4,74,44),(5,45,75),(6,76,46),(7,47,77),(8,78,48),(9,33,79),(10,80,34),(11,35,65),(12,66,36),(13,37,67),(14,68,38),(15,39,69),(16,70,40),(17,62,90),(18,91,63),(19,64,92),(20,93,49),(21,50,94),(22,95,51),(23,52,96),(24,81,53),(25,54,82),(26,83,55),(27,56,84),(28,85,57),(29,58,86),(30,87,59),(31,60,88),(32,89,61)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,82),(26,81),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,80),(49,59),(50,58),(51,57),(52,56),(53,55),(60,64),(61,63)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | C3⋊D4 | C3⋊D4 | D16 | D4⋊S3 | D4⋊S3 | C3⋊D16 |
| kernel | C2×C3⋊D16 | C2×C3⋊C16 | C3⋊D16 | C2×D24 | C6×D8 | C2×D8 | C24 | C2×C12 | C2×C8 | D8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C2×C3⋊D16 ►in GL5(𝔽97)
| 96 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 96 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 28 | 45 |
| 0 | 0 | 0 | 26 | 73 |
| 96 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 96 |
G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,96,1,0,0,0,96,0,0,0,0,0,0,1,0,0,0,0,0,1],[96,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,28,26,0,0,0,45,73],[96,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,1,1,0,0,0,0,96] >;
C2×C3⋊D16 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes D_{16} % in TeX
G:=Group("C2xC3:D16"); // GroupNames label
G:=SmallGroup(192,705);
// by ID
G=gap.SmallGroup(192,705);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,675,185,192,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations