metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.5D8, C2.D8⋊2S3, D6⋊C8⋊15C2, C4⋊C4.47D6, C6.29(C2×D8), (C2×C8).27D6, C2.13(S3×D8), C12⋊D4.7C2, C6.Q16⋊21C2, C6.D8⋊20C2, C2.D24⋊14C2, C4.80(C4○D12), C12.37(C4○D4), (C2×Dic3).48D4, (C22×S3).85D4, C22.228(S3×D4), C3⋊3(C22.D8), (C2×C24).169C22, (C2×C12).298C23, C4.28(Q8⋊3S3), C2.22(Q16⋊S3), (C2×D12).81C22, C6.70(C8.C22), C2.15(D6.D4), C4⋊Dic3.124C22, C6.45(C22.D4), (S3×C4⋊C4)⋊7C2, (C3×C2.D8)⋊11C2, (C2×C6).303(C2×D4), (C2×C3⋊C8).69C22, (S3×C2×C4).38C22, (C3×C4⋊C4).91C22, (C2×C4).401(C22×S3), SmallGroup(192,441)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.5D8
G = < a,b,c,d | a6=b2=c8=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 400 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22⋊C8, D4⋊C4, C2.D8, C2.D8, C2×C4⋊C4, C4⋊D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C22.D8, C6.Q16, C6.D8, D6⋊C8, C2.D24, C3×C2.D8, S3×C4⋊C4, C12⋊D4, D6.5D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C22×S3, C22.D4, C2×D8, C8.C22, C4○D12, S3×D4, Q8⋊3S3, C22.D8, D6.D4, S3×D8, Q16⋊S3, D6.5D8
►On 96 points
(1 21 45 77 49 82)(2 22 46 78 50 83)(3 23 47 79 51 84)(4 24 48 80 52 85)(5 17 41 73 53 86)(6 18 42 74 54 87)(7 19 43 75 55 88)(8 20 44 76 56 81)(9 26 71 57 95 35)(10 27 72 58 96 36)(11 28 65 59 89 37)(12 29 66 60 90 38)(13 30 67 61 91 39)(14 31 68 62 92 40)(15 32 69 63 93 33)(16 25 70 64 94 34)
(1 25)(2 95)(3 27)(4 89)(5 29)(6 91)(7 31)(8 93)(9 50)(10 23)(11 52)(12 17)(13 54)(14 19)(15 56)(16 21)(18 61)(20 63)(22 57)(24 59)(26 78)(28 80)(30 74)(32 76)(33 81)(34 45)(35 83)(36 47)(37 85)(38 41)(39 87)(40 43)(42 67)(44 69)(46 71)(48 65)(49 64)(51 58)(53 60)(55 62)(66 86)(68 88)(70 82)(72 84)(73 90)(75 92)(77 94)(79 96)
(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 36 77 72)(2 35 78 71)(3 34 79 70)(4 33 80 69)(5 40 73 68)(6 39 74 67)(7 38 75 66)(8 37 76 65)(9 50 57 22)(10 49 58 21)(11 56 59 20)(12 55 60 19)(13 54 61 18)(14 53 62 17)(15 52 63 24)(16 51 64 23)(25 84 94 47)(26 83 95 46)(27 82 96 45)(28 81 89 44)(29 88 90 43)(30 87 91 42)(31 86 92 41)(32 85 93 48)
G:=sub<Sym(96)| (1,21,45,77,49,82)(2,22,46,78,50,83)(3,23,47,79,51,84)(4,24,48,80,52,85)(5,17,41,73,53,86)(6,18,42,74,54,87)(7,19,43,75,55,88)(8,20,44,76,56,81)(9,26,71,57,95,35)(10,27,72,58,96,36)(11,28,65,59,89,37)(12,29,66,60,90,38)(13,30,67,61,91,39)(14,31,68,62,92,40)(15,32,69,63,93,33)(16,25,70,64,94,34), (1,25)(2,95)(3,27)(4,89)(5,29)(6,91)(7,31)(8,93)(9,50)(10,23)(11,52)(12,17)(13,54)(14,19)(15,56)(16,21)(18,61)(20,63)(22,57)(24,59)(26,78)(28,80)(30,74)(32,76)(33,81)(34,45)(35,83)(36,47)(37,85)(38,41)(39,87)(40,43)(42,67)(44,69)(46,71)(48,65)(49,64)(51,58)(53,60)(55,62)(66,86)(68,88)(70,82)(72,84)(73,90)(75,92)(77,94)(79,96), (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), (1,36,77,72)(2,35,78,71)(3,34,79,70)(4,33,80,69)(5,40,73,68)(6,39,74,67)(7,38,75,66)(8,37,76,65)(9,50,57,22)(10,49,58,21)(11,56,59,20)(12,55,60,19)(13,54,61,18)(14,53,62,17)(15,52,63,24)(16,51,64,23)(25,84,94,47)(26,83,95,46)(27,82,96,45)(28,81,89,44)(29,88,90,43)(30,87,91,42)(31,86,92,41)(32,85,93,48)>;
G:=Group( (1,21,45,77,49,82)(2,22,46,78,50,83)(3,23,47,79,51,84)(4,24,48,80,52,85)(5,17,41,73,53,86)(6,18,42,74,54,87)(7,19,43,75,55,88)(8,20,44,76,56,81)(9,26,71,57,95,35)(10,27,72,58,96,36)(11,28,65,59,89,37)(12,29,66,60,90,38)(13,30,67,61,91,39)(14,31,68,62,92,40)(15,32,69,63,93,33)(16,25,70,64,94,34), (1,25)(2,95)(3,27)(4,89)(5,29)(6,91)(7,31)(8,93)(9,50)(10,23)(11,52)(12,17)(13,54)(14,19)(15,56)(16,21)(18,61)(20,63)(22,57)(24,59)(26,78)(28,80)(30,74)(32,76)(33,81)(34,45)(35,83)(36,47)(37,85)(38,41)(39,87)(40,43)(42,67)(44,69)(46,71)(48,65)(49,64)(51,58)(53,60)(55,62)(66,86)(68,88)(70,82)(72,84)(73,90)(75,92)(77,94)(79,96), (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), (1,36,77,72)(2,35,78,71)(3,34,79,70)(4,33,80,69)(5,40,73,68)(6,39,74,67)(7,38,75,66)(8,37,76,65)(9,50,57,22)(10,49,58,21)(11,56,59,20)(12,55,60,19)(13,54,61,18)(14,53,62,17)(15,52,63,24)(16,51,64,23)(25,84,94,47)(26,83,95,46)(27,82,96,45)(28,81,89,44)(29,88,90,43)(30,87,91,42)(31,86,92,41)(32,85,93,48) );
G=PermutationGroup([[(1,21,45,77,49,82),(2,22,46,78,50,83),(3,23,47,79,51,84),(4,24,48,80,52,85),(5,17,41,73,53,86),(6,18,42,74,54,87),(7,19,43,75,55,88),(8,20,44,76,56,81),(9,26,71,57,95,35),(10,27,72,58,96,36),(11,28,65,59,89,37),(12,29,66,60,90,38),(13,30,67,61,91,39),(14,31,68,62,92,40),(15,32,69,63,93,33),(16,25,70,64,94,34)], [(1,25),(2,95),(3,27),(4,89),(5,29),(6,91),(7,31),(8,93),(9,50),(10,23),(11,52),(12,17),(13,54),(14,19),(15,56),(16,21),(18,61),(20,63),(22,57),(24,59),(26,78),(28,80),(30,74),(32,76),(33,81),(34,45),(35,83),(36,47),(37,85),(38,41),(39,87),(40,43),(42,67),(44,69),(46,71),(48,65),(49,64),(51,58),(53,60),(55,62),(66,86),(68,88),(70,82),(72,84),(73,90),(75,92),(77,94),(79,96)], [(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)], [(1,36,77,72),(2,35,78,71),(3,34,79,70),(4,33,80,69),(5,40,73,68),(6,39,74,67),(7,38,75,66),(8,37,76,65),(9,50,57,22),(10,49,58,21),(11,56,59,20),(12,55,60,19),(13,54,61,18),(14,53,62,17),(15,52,63,24),(16,51,64,23),(25,84,94,47),(26,83,95,46),(27,82,96,45),(28,81,89,44),(29,88,90,43),(30,87,91,42),(31,86,92,41),(32,85,93,48)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D8 | C4○D12 | C8.C22 | Q8⋊3S3 | S3×D4 | S3×D8 | Q16⋊S3 |
| kernel | D6.5D8 | C6.Q16 | C6.D8 | D6⋊C8 | C2.D24 | C3×C2.D8 | S3×C4⋊C4 | C12⋊D4 | C2.D8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | D6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6.5D8 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 66 | 7 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 32 | 32 |
| 0 | 0 | 57 | 0 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[66,14,0,0,7,7,0,0,0,0,1,0,0,0,0,1],[43,60,0,0,13,30,0,0,0,0,32,57,0,0,32,0],[7,59,0,0,14,66,0,0,0,0,72,0,0,0,71,1] >;
D6.5D8 in GAP, Magma, Sage, TeX
D_6._5D_8
% in TeX
G:=Group("D6.5D8"); // GroupNames label
G:=SmallGroup(192,441);
// by ID
G=gap.SmallGroup(192,441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,254,219,268,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations