metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C16⋊1D6, D48⋊2C2, C8.3D12, C24.2D4, C48⋊1C22, C4.14D24, C12.14D8, D24⋊8C22, M5(2)⋊1S3, C22.5D24, C24.59C23, Dic12⋊7C22, (C2×C6).6D8, C4○D24⋊9C2, C48⋊C2⋊1C2, C6.13(C2×D8), (C2×C8).73D6, (C2×D24)⋊11C2, C3⋊1(C16⋊C22), (C2×C4).41D12, C2.15(C2×D24), C4.40(C2×D12), (C2×C12).128D4, C12.283(C2×D4), (C3×M5(2))⋊1C2, C8.49(C22×S3), (C2×C24).59C22, SmallGroup(192,467)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊D6
G = < a,b,c | a16=b6=c2=1, bab-1=a9, cac=a7, cbc=b-1 >
Subgroups: 424 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C16, C2×C8, D8, SD16, Q16, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C22×S3, M5(2), D16, SD32, C2×D8, C4○D8, C48, C24⋊C2, D24, D24, D24, Dic12, C2×C24, C2×D12, C4○D12, C16⋊C22, D48, C48⋊C2, C3×M5(2), C2×D24, C4○D24, C16⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C2×D8, D24, C2×D12, C16⋊C22, C2×D24, C16⋊D6
►On 48 points
(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)
(1 46 27)(2 39 28 10 47 20)(3 48 29)(4 41 30 12 33 22)(5 34 31)(6 43 32 14 35 24)(7 36 17)(8 45 18 16 37 26)(9 38 19)(11 40 21)(13 42 23)(15 44 25)
(1 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)
G:=sub<Sym(48)| (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), (1,46,27)(2,39,28,10,47,20)(3,48,29)(4,41,30,12,33,22)(5,34,31)(6,43,32,14,35,24)(7,36,17)(8,45,18,16,37,26)(9,38,19)(11,40,21)(13,42,23)(15,44,25), (1,27)(2,18)(3,25)(4,32)(5,23)(6,30)(7,21)(8,28)(9,19)(10,26)(11,17)(12,24)(13,31)(14,22)(15,29)(16,20)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)>;
G:=Group( (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), (1,46,27)(2,39,28,10,47,20)(3,48,29)(4,41,30,12,33,22)(5,34,31)(6,43,32,14,35,24)(7,36,17)(8,45,18,16,37,26)(9,38,19)(11,40,21)(13,42,23)(15,44,25), (1,27)(2,18)(3,25)(4,32)(5,23)(6,30)(7,21)(8,28)(9,19)(10,26)(11,17)(12,24)(13,31)(14,22)(15,29)(16,20)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48) );
G=PermutationGroup([[(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)], [(1,46,27),(2,39,28,10,47,20),(3,48,29),(4,41,30,12,33,22),(5,34,31),(6,43,32,14,35,24),(7,36,17),(8,45,18,16,37,26),(9,38,19),(11,40,21),(13,42,23),(15,44,25)], [(1,27),(2,18),(3,25),(4,32),(5,23),(6,30),(7,21),(8,28),(9,19),(10,26),(11,17),(12,24),(13,31),(14,22),(15,29),(16,20),(33,35),(34,42),(36,40),(37,47),(39,45),(41,43),(44,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 8C | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 24 | 24 | 24 | 2 | 2 | 2 | 24 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | D12 | D12 | D24 | D24 | C16⋊C22 | C16⋊D6 |
| kernel | C16⋊D6 | D48 | C48⋊C2 | C3×M5(2) | C2×D24 | C4○D24 | M5(2) | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C16⋊D6 ►in GL6(𝔽97)
| 2 | 79 | 0 | 0 | 0 | 0 |
| 18 | 81 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 90 | 90 | 0 | 0 |
| 0 | 0 | 7 | 90 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 96 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 |
| 96 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 90 |
| 0 | 0 | 0 | 0 | 90 | 90 |
G:=sub<GL(6,GF(97))| [2,18,0,0,0,0,79,81,0,0,0,0,0,0,0,0,90,7,0,0,0,0,90,90,0,0,1,0,0,0,0,0,0,1,0,0],[0,96,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,96],[96,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,7,90,0,0,0,0,90,90] >;
C16⋊D6 in GAP, Magma, Sage, TeX
C_{16}\rtimes D_6 % in TeX
G:=Group("C16:D6"); // GroupNames label
G:=SmallGroup(192,467);
// by ID
G=gap.SmallGroup(192,467);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,142,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^16=b^6=c^2=1,b*a*b^-1=a^9,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations