metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D6.C8, C48⋊6C2, C16⋊3S3, C8.20D6, Dic3.C8, C3⋊1M5(2), C24.24C22, C3⋊C16⋊4C2, C3⋊C8.2C4, C2.3(S3×C8), C6.2(C2×C8), (S3×C8).2C2, (C4×S3).2C4, C4.17(C4×S3), C12.22(C2×C4), SmallGroup(96,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.C8
G = < a,b,c | a6=b2=1, c8=a3, bab=a-1, ac=ca, cbc-1=a3b >
Smallest permutation representation of D6.C8
►On 48 points
(1 48 31 9 40 23)(2 33 32 10 41 24)(3 34 17 11 42 25)(4 35 18 12 43 26)(5 36 19 13 44 27)(6 37 20 14 45 28)(7 38 21 15 46 29)(8 39 22 16 47 30)
(1 23)(2 32)(3 25)(4 18)(5 27)(6 20)(7 29)(8 22)(9 31)(10 24)(11 17)(12 26)(13 19)(14 28)(15 21)(16 30)(34 42)(36 44)(38 46)(40 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)
G:=sub<Sym(48)| (1,48,31,9,40,23)(2,33,32,10,41,24)(3,34,17,11,42,25)(4,35,18,12,43,26)(5,36,19,13,44,27)(6,37,20,14,45,28)(7,38,21,15,46,29)(8,39,22,16,47,30), (1,23)(2,32)(3,25)(4,18)(5,27)(6,20)(7,29)(8,22)(9,31)(10,24)(11,17)(12,26)(13,19)(14,28)(15,21)(16,30)(34,42)(36,44)(38,46)(40,48), (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)>;
G:=Group( (1,48,31,9,40,23)(2,33,32,10,41,24)(3,34,17,11,42,25)(4,35,18,12,43,26)(5,36,19,13,44,27)(6,37,20,14,45,28)(7,38,21,15,46,29)(8,39,22,16,47,30), (1,23)(2,32)(3,25)(4,18)(5,27)(6,20)(7,29)(8,22)(9,31)(10,24)(11,17)(12,26)(13,19)(14,28)(15,21)(16,30)(34,42)(36,44)(38,46)(40,48), (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) );
G=PermutationGroup([[(1,48,31,9,40,23),(2,33,32,10,41,24),(3,34,17,11,42,25),(4,35,18,12,43,26),(5,36,19,13,44,27),(6,37,20,14,45,28),(7,38,21,15,46,29),(8,39,22,16,47,30)], [(1,23),(2,32),(3,25),(4,18),(5,27),(6,20),(7,29),(8,22),(9,31),(10,24),(11,17),(12,26),(13,19),(14,28),(15,21),(16,30),(34,42),(36,44),(38,46),(40,48)], [(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)]])
D6.C8 is a maximal subgroup of
D12.4C8 S3×M5(2) C16.12D6 D8⋊D6 D48⋊C2 SD32⋊S3 Q32⋊S3 C16⋊D9 C24.61D6 C24.62D6 C48⋊S3 C40.52D6 D30.5C8 C80⋊S3 C15⋊M5(2) D30.C8
D6.C8 is a maximal quotient of
Dic3⋊C16 C48⋊10C4 D6⋊C16 C16⋊D9 C24.61D6 C24.62D6 C48⋊S3 C40.52D6 D30.5C8 C80⋊S3 C15⋊M5(2) D30.C8
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 24C | 24D | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D6 | C4×S3 | M5(2) | S3×C8 | D6.C8 |
| kernel | D6.C8 | C3⋊C16 | C48 | S3×C8 | C3⋊C8 | C4×S3 | Dic3 | D6 | C16 | C8 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of D6.C8 ►in GL2(𝔽41) generated by
| 40 | 38 |
| 1 | 2 |
| 40 | 0 |
| 1 | 1 |
| 3 | 6 |
| 39 | 38 |
G:=sub<GL(2,GF(41))| [40,1,38,2],[40,1,0,1],[3,39,6,38] >;
D6.C8 in GAP, Magma, Sage, TeX
D_6.C_8
% in TeX
G:=Group("D6.C8"); // GroupNames label
G:=SmallGroup(96,5);
// by ID
G=gap.SmallGroup(96,5);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,31,50,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^6=b^2=1,c^8=a^3,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations
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