metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊5C4, C6.7D16, C24.5D4, C8.15D12, C6.7SD32, C12.4SD16, C2.D8⋊1S3, C8.12(C4×S3), C24.9(C2×C4), (C2×C6).33D8, C3⋊1(C2.D16), C4.2(D6⋊C4), (C2×D24).8C2, (C2×C12).91D4, (C2×C8).221D6, C2.2(C3⋊D16), C6.5(D4⋊C4), C12.2(C22⋊C4), (C2×C24).73C22, C4.1(Q8⋊2S3), C2.2(C8.6D6), C2.7(C6.D8), C22.14(D4⋊S3), (C2×C3⋊C16)⋊4C2, (C3×C2.D8)⋊1C2, (C2×C4).115(C3⋊D4), SmallGroup(192,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D16
G = < a,b,c | a6=b16=c2=1, bab-1=cac=a-1, cbc=a3b-1 >
Subgroups: 304 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, C12, C12, D6, C2×C6, C16, C4⋊C4, C2×C8, D8, C2×D4, C24, D12, C2×C12, C2×C12, C22×S3, C2.D8, C2×C16, C2×D8, C3⋊C16, D24, D24, C3×C4⋊C4, C2×C24, C2×D12, C2.D16, C2×C3⋊C16, C3×C2.D8, C2×D24, C6.D16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D16, SD32, D6⋊C4, D4⋊S3, Q8⋊2S3, C2.D16, C6.D8, C3⋊D16, C8.6D6, C6.D16
►On 96 points
(1 28 83 71 34 53)(2 54 35 72 84 29)(3 30 85 73 36 55)(4 56 37 74 86 31)(5 32 87 75 38 57)(6 58 39 76 88 17)(7 18 89 77 40 59)(8 60 41 78 90 19)(9 20 91 79 42 61)(10 62 43 80 92 21)(11 22 93 65 44 63)(12 64 45 66 94 23)(13 24 95 67 46 49)(14 50 47 68 96 25)(15 26 81 69 48 51)(16 52 33 70 82 27)
(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 70)(3 15)(4 68)(5 13)(6 66)(7 11)(8 80)(10 78)(12 76)(14 74)(16 72)(17 94)(18 63)(19 92)(20 61)(21 90)(22 59)(23 88)(24 57)(25 86)(26 55)(27 84)(28 53)(29 82)(30 51)(31 96)(32 49)(33 54)(34 83)(35 52)(36 81)(37 50)(38 95)(39 64)(40 93)(41 62)(42 91)(43 60)(44 89)(45 58)(46 87)(47 56)(48 85)(65 77)(67 75)(69 73)
G:=sub<Sym(96)| (1,28,83,71,34,53)(2,54,35,72,84,29)(3,30,85,73,36,55)(4,56,37,74,86,31)(5,32,87,75,38,57)(6,58,39,76,88,17)(7,18,89,77,40,59)(8,60,41,78,90,19)(9,20,91,79,42,61)(10,62,43,80,92,21)(11,22,93,65,44,63)(12,64,45,66,94,23)(13,24,95,67,46,49)(14,50,47,68,96,25)(15,26,81,69,48,51)(16,52,33,70,82,27), (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), (2,70)(3,15)(4,68)(5,13)(6,66)(7,11)(8,80)(10,78)(12,76)(14,74)(16,72)(17,94)(18,63)(19,92)(20,61)(21,90)(22,59)(23,88)(24,57)(25,86)(26,55)(27,84)(28,53)(29,82)(30,51)(31,96)(32,49)(33,54)(34,83)(35,52)(36,81)(37,50)(38,95)(39,64)(40,93)(41,62)(42,91)(43,60)(44,89)(45,58)(46,87)(47,56)(48,85)(65,77)(67,75)(69,73)>;
G:=Group( (1,28,83,71,34,53)(2,54,35,72,84,29)(3,30,85,73,36,55)(4,56,37,74,86,31)(5,32,87,75,38,57)(6,58,39,76,88,17)(7,18,89,77,40,59)(8,60,41,78,90,19)(9,20,91,79,42,61)(10,62,43,80,92,21)(11,22,93,65,44,63)(12,64,45,66,94,23)(13,24,95,67,46,49)(14,50,47,68,96,25)(15,26,81,69,48,51)(16,52,33,70,82,27), (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), (2,70)(3,15)(4,68)(5,13)(6,66)(7,11)(8,80)(10,78)(12,76)(14,74)(16,72)(17,94)(18,63)(19,92)(20,61)(21,90)(22,59)(23,88)(24,57)(25,86)(26,55)(27,84)(28,53)(29,82)(30,51)(31,96)(32,49)(33,54)(34,83)(35,52)(36,81)(37,50)(38,95)(39,64)(40,93)(41,62)(42,91)(43,60)(44,89)(45,58)(46,87)(47,56)(48,85)(65,77)(67,75)(69,73) );
G=PermutationGroup([[(1,28,83,71,34,53),(2,54,35,72,84,29),(3,30,85,73,36,55),(4,56,37,74,86,31),(5,32,87,75,38,57),(6,58,39,76,88,17),(7,18,89,77,40,59),(8,60,41,78,90,19),(9,20,91,79,42,61),(10,62,43,80,92,21),(11,22,93,65,44,63),(12,64,45,66,94,23),(13,24,95,67,46,49),(14,50,47,68,96,25),(15,26,81,69,48,51),(16,52,33,70,82,27)], [(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)], [(2,70),(3,15),(4,68),(5,13),(6,66),(7,11),(8,80),(10,78),(12,76),(14,74),(16,72),(17,94),(18,63),(19,92),(20,61),(21,90),(22,59),(23,88),(24,57),(25,86),(26,55),(27,84),(28,53),(29,82),(30,51),(31,96),(32,49),(33,54),(34,83),(35,52),(36,81),(37,50),(38,95),(39,64),(40,93),(41,62),(42,91),(43,60),(44,89),(45,58),(46,87),(47,56),(48,85),(65,77),(67,75),(69,73)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | SD16 | D8 | C4×S3 | D12 | C3⋊D4 | D16 | SD32 | Q8⋊2S3 | D4⋊S3 | C3⋊D16 | C8.6D6 |
| kernel | C6.D16 | C2×C3⋊C16 | C3×C2.D8 | C2×D24 | D24 | C2.D8 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C6.D16 ►in GL5(𝔽97)
| 96 | 0 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 96 |
| 75 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 54 | 35 |
| 0 | 0 | 0 | 3 | 63 |
| 96 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 64 | 96 |
G:=sub<GL(5,GF(97))| [96,0,0,0,0,0,96,1,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,96],[75,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,54,3,0,0,0,35,63],[96,0,0,0,0,0,1,96,0,0,0,0,96,0,0,0,0,0,1,64,0,0,0,0,96] >;
C6.D16 in GAP, Magma, Sage, TeX
C_6.D_{16} % in TeX
G:=Group("C6.D16"); // GroupNames label
G:=SmallGroup(192,50);
// by ID
G=gap.SmallGroup(192,50);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,675,346,192,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^6=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^-1>;
// generators/relations