direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C2.D16, D8⋊1C12, C24.88D4, C6.13D16, C6.9SD32, C12.30SD16, (C2×C48)⋊7C2, (C2×C16)⋊3C6, (C3×D8)⋊7C4, C2.D8⋊1C6, C8.7(C2×C12), (C6×D8).7C2, (C2×D8).1C6, C2.1(C3×D16), C8.14(C3×D4), (C2×C6).49D8, C24.53(C2×C4), C4.1(C3×SD16), C2.1(C3×SD32), C22.8(C3×D8), (C2×C12).405D4, C6.37(D4⋊C4), C12.69(C22⋊C4), (C2×C24).394C22, (C2×C8).71(C2×C6), (C3×C2.D8)⋊10C2, (C2×C4).59(C3×D4), C4.1(C3×C22⋊C4), C2.6(C3×D4⋊C4), SmallGroup(192,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C2.D16
G = < a,b,c,d | a3=b2=c16=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 178 in 66 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C16, C4⋊C4, C2×C8, D8, D8, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C2.D8, C2×C16, C2×D8, C48, C3×C4⋊C4, C2×C24, C3×D8, C3×D8, C6×D4, C2.D16, C3×C2.D8, C2×C48, C6×D8, C3×C2.D16
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, C2×C12, C3×D4, D4⋊C4, D16, SD32, C3×C22⋊C4, C3×D8, C3×SD16, C2.D16, C3×D4⋊C4, C3×D16, C3×SD32, C3×C2.D16
►On 96 points
Generators in S96
(1 22 84)(2 23 85)(3 24 86)(4 25 87)(5 26 88)(6 27 89)(7 28 90)(8 29 91)(9 30 92)(10 31 93)(11 32 94)(12 17 95)(13 18 96)(14 19 81)(15 20 82)(16 21 83)(33 72 55)(34 73 56)(35 74 57)(36 75 58)(37 76 59)(38 77 60)(39 78 61)(40 79 62)(41 80 63)(42 65 64)(43 66 49)(44 67 50)(45 68 51)(46 69 52)(47 70 53)(48 71 54)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 41)(17 76)(18 77)(19 78)(20 79)(21 80)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 81)(62 82)(63 83)(64 84)
(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 16 42 41)(2 40 43 15)(3 14 44 39)(4 38 45 13)(5 12 46 37)(6 36 47 11)(7 10 48 35)(8 34 33 9)(17 69 76 26)(18 25 77 68)(19 67 78 24)(20 23 79 66)(21 65 80 22)(27 75 70 32)(28 31 71 74)(29 73 72 30)(49 82 85 62)(50 61 86 81)(51 96 87 60)(52 59 88 95)(53 94 89 58)(54 57 90 93)(55 92 91 56)(63 84 83 64)
G:=sub<Sym(96)| (1,22,84)(2,23,85)(3,24,86)(4,25,87)(5,26,88)(6,27,89)(7,28,90)(8,29,91)(9,30,92)(10,31,93)(11,32,94)(12,17,95)(13,18,96)(14,19,81)(15,20,82)(16,21,83)(33,72,55)(34,73,56)(35,74,57)(36,75,58)(37,76,59)(38,77,60)(39,78,61)(40,79,62)(41,80,63)(42,65,64)(43,66,49)(44,67,50)(45,68,51)(46,69,52)(47,70,53)(48,71,54), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,76)(18,77)(19,78)(20,79)(21,80)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,81)(62,82)(63,83)(64,84), (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), (1,16,42,41)(2,40,43,15)(3,14,44,39)(4,38,45,13)(5,12,46,37)(6,36,47,11)(7,10,48,35)(8,34,33,9)(17,69,76,26)(18,25,77,68)(19,67,78,24)(20,23,79,66)(21,65,80,22)(27,75,70,32)(28,31,71,74)(29,73,72,30)(49,82,85,62)(50,61,86,81)(51,96,87,60)(52,59,88,95)(53,94,89,58)(54,57,90,93)(55,92,91,56)(63,84,83,64)>;
G:=Group( (1,22,84)(2,23,85)(3,24,86)(4,25,87)(5,26,88)(6,27,89)(7,28,90)(8,29,91)(9,30,92)(10,31,93)(11,32,94)(12,17,95)(13,18,96)(14,19,81)(15,20,82)(16,21,83)(33,72,55)(34,73,56)(35,74,57)(36,75,58)(37,76,59)(38,77,60)(39,78,61)(40,79,62)(41,80,63)(42,65,64)(43,66,49)(44,67,50)(45,68,51)(46,69,52)(47,70,53)(48,71,54), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,41)(17,76)(18,77)(19,78)(20,79)(21,80)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,81)(62,82)(63,83)(64,84), (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), (1,16,42,41)(2,40,43,15)(3,14,44,39)(4,38,45,13)(5,12,46,37)(6,36,47,11)(7,10,48,35)(8,34,33,9)(17,69,76,26)(18,25,77,68)(19,67,78,24)(20,23,79,66)(21,65,80,22)(27,75,70,32)(28,31,71,74)(29,73,72,30)(49,82,85,62)(50,61,86,81)(51,96,87,60)(52,59,88,95)(53,94,89,58)(54,57,90,93)(55,92,91,56)(63,84,83,64) );
G=PermutationGroup([[(1,22,84),(2,23,85),(3,24,86),(4,25,87),(5,26,88),(6,27,89),(7,28,90),(8,29,91),(9,30,92),(10,31,93),(11,32,94),(12,17,95),(13,18,96),(14,19,81),(15,20,82),(16,21,83),(33,72,55),(34,73,56),(35,74,57),(36,75,58),(37,76,59),(38,77,60),(39,78,61),(40,79,62),(41,80,63),(42,65,64),(43,66,49),(44,67,50),(45,68,51),(46,69,52),(47,70,53),(48,71,54)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,41),(17,76),(18,77),(19,78),(20,79),(21,80),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,81),(62,82),(63,83),(64,84)], [(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)], [(1,16,42,41),(2,40,43,15),(3,14,44,39),(4,38,45,13),(5,12,46,37),(6,36,47,11),(7,10,48,35),(8,34,33,9),(17,69,76,26),(18,25,77,68),(19,67,78,24),(20,23,79,66),(21,65,80,22),(27,75,70,32),(28,31,71,74),(29,73,72,30),(49,82,85,62),(50,61,86,81),(51,96,87,60),(52,59,88,95),(53,94,89,58),(54,57,90,93),(55,92,91,56),(63,84,83,64)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | SD16 | D8 | C3×D4 | C3×D4 | D16 | SD32 | C3×SD16 | C3×D8 | C3×D16 | C3×SD32 |
| kernel | C3×C2.D16 | C3×C2.D8 | C2×C48 | C6×D8 | C2.D16 | C3×D8 | C2.D8 | C2×C16 | C2×D8 | D8 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C2.D16 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 96 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 75 | 0 | 0 |
| 0 | 2 | 26 |
| 0 | 71 | 2 |
| 75 | 0 | 0 |
| 0 | 71 | 2 |
| 0 | 2 | 26 |
G:=sub<GL(3,GF(97))| [1,0,0,0,61,0,0,0,61],[96,0,0,0,1,0,0,0,1],[75,0,0,0,2,71,0,26,2],[75,0,0,0,71,2,0,2,26] >;
C3×C2.D16 in GAP, Magma, Sage, TeX
C_3\times C_2.D_{16} % in TeX
G:=Group("C3xC2.D16"); // GroupNames label
G:=SmallGroup(192,163);
// by ID
G=gap.SmallGroup(192,163);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,850,360,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^16=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations