direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C6×D16, C12.44D8, C24.68D4, C48⋊10C22, C24.63C23, (C2×C16)⋊5C6, C16⋊2(C2×C6), (C2×D8)⋊6C6, D8⋊1(C2×C6), C8.9(C3×D4), C4.6(C3×D8), C4.7(C6×D4), (C2×C48)⋊12C2, (C6×D8)⋊20C2, C6.84(C2×D8), C2.12(C6×D8), (C2×C6).55D8, C8.3(C22×C6), C12.314(C2×D4), (C2×C12).426D4, (C3×D8)⋊17C22, C22.14(C3×D8), (C2×C24).404C22, (C2×C8).84(C2×C6), (C2×C4).82(C3×D4), SmallGroup(192,938)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D16
G = < a,b,c | a6=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 274 in 98 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, D4, C23, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, C2×D4, C24, C2×C12, C3×D4, C22×C6, C2×C16, D16, C2×D8, C48, C2×C24, C3×D8, C3×D8, C6×D4, C2×D16, C2×C48, C3×D16, C6×D8, C6×D16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, D16, C2×D8, C3×D8, C6×D4, C2×D16, C3×D16, C6×D8, C6×D16
►On 96 points
Generators in S96
(1 79 35 61 21 89)(2 80 36 62 22 90)(3 65 37 63 23 91)(4 66 38 64 24 92)(5 67 39 49 25 93)(6 68 40 50 26 94)(7 69 41 51 27 95)(8 70 42 52 28 96)(9 71 43 53 29 81)(10 72 44 54 30 82)(11 73 45 55 31 83)(12 74 46 56 32 84)(13 75 47 57 17 85)(14 76 48 58 18 86)(15 77 33 59 19 87)(16 78 34 60 20 88)
(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 52)(2 51)(3 50)(4 49)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 82)(34 81)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)
G:=sub<Sym(96)| (1,79,35,61,21,89)(2,80,36,62,22,90)(3,65,37,63,23,91)(4,66,38,64,24,92)(5,67,39,49,25,93)(6,68,40,50,26,94)(7,69,41,51,27,95)(8,70,42,52,28,96)(9,71,43,53,29,81)(10,72,44,54,30,82)(11,73,45,55,31,83)(12,74,46,56,32,84)(13,75,47,57,17,85)(14,76,48,58,18,86)(15,77,33,59,19,87)(16,78,34,60,20,88), (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), (1,52)(2,51)(3,50)(4,49)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,82)(34,81)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)>;
G:=Group( (1,79,35,61,21,89)(2,80,36,62,22,90)(3,65,37,63,23,91)(4,66,38,64,24,92)(5,67,39,49,25,93)(6,68,40,50,26,94)(7,69,41,51,27,95)(8,70,42,52,28,96)(9,71,43,53,29,81)(10,72,44,54,30,82)(11,73,45,55,31,83)(12,74,46,56,32,84)(13,75,47,57,17,85)(14,76,48,58,18,86)(15,77,33,59,19,87)(16,78,34,60,20,88), (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), (1,52)(2,51)(3,50)(4,49)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,82)(34,81)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83) );
G=PermutationGroup([[(1,79,35,61,21,89),(2,80,36,62,22,90),(3,65,37,63,23,91),(4,66,38,64,24,92),(5,67,39,49,25,93),(6,68,40,50,26,94),(7,69,41,51,27,95),(8,70,42,52,28,96),(9,71,43,53,29,81),(10,72,44,54,30,82),(11,73,45,55,31,83),(12,74,46,56,32,84),(13,75,47,57,17,85),(14,76,48,58,18,86),(15,77,33,59,19,87),(16,78,34,60,20,88)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,82),(34,81),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | D8 | D8 | C3×D4 | C3×D4 | D16 | C3×D8 | C3×D8 | C3×D16 |
| kernel | C6×D16 | C2×C48 | C3×D16 | C6×D8 | C2×D16 | C2×C16 | D16 | C2×D8 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C6×D16 ►in GL4(𝔽97) generated by
| 96 | 0 | 0 | 0 |
| 0 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 96 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 26 | 95 |
| 0 | 0 | 2 | 26 |
| 96 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 71 | 2 |
| 0 | 0 | 2 | 26 |
G:=sub<GL(4,GF(97))| [96,0,0,0,0,35,0,0,0,0,1,0,0,0,0,1],[96,0,0,0,0,96,0,0,0,0,26,2,0,0,95,26],[96,0,0,0,0,1,0,0,0,0,71,2,0,0,2,26] >;
C6×D16 in GAP, Magma, Sage, TeX
C_6\times D_{16} % in TeX
G:=Group("C6xD16"); // GroupNames label
G:=SmallGroup(192,938);
// by ID
G=gap.SmallGroup(192,938);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,2524,1271,242,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c|a^6=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations