metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊6D14, SD16⋊4D14, D28.42D4, C56.3C23, C28.22C24, M4(2)⋊10D14, Dic14.42D4, Dic28⋊2C22, D28.15C23, Dic14.15C23, C4○D4⋊4D14, C8⋊C22⋊5D7, C7⋊D4.5D4, D8⋊D7⋊4C2, D8⋊3D7⋊2C2, D28.C4⋊2C2, (D7×SD16)⋊2C2, C4.116(D4×D7), C7⋊4(D4○SD16), (C8×D7)⋊4C22, (C7×D8)⋊4C22, C7⋊C8.26C23, D4⋊6D14⋊8C2, (Q8×D7)⋊3C22, C8.3(C22×D7), D4⋊D7⋊15C22, D14.33(C2×D4), C8.D14⋊2C2, C28.243(C2×D4), SD16⋊D7⋊2C2, C56⋊C2⋊4C22, C8⋊D7⋊4C22, Q8⋊D7⋊14C22, (D4×D7).3C22, C22.15(D4×D7), C4.22(C23×D7), (C2×D4).117D14, D4.8D14⋊4C2, D4⋊2D7⋊4C22, D4.D7⋊14C22, Dic7.38(C2×D4), (C7×SD16)⋊4C22, D4.15(C22×D7), (C4×D7).14C23, (C7×D4).15C23, C7⋊Q16⋊13C22, D4.10D14⋊7C2, Q8.15(C22×D7), (C7×Q8).15C23, (C2×C28).113C23, C4○D28.29C22, C14.123(C22×D4), (C7×M4(2))⋊4C22, (C2×Dic14)⋊40C22, (D4×C14).168C22, C2.96(C2×D4×D7), (C7×C8⋊C22)⋊4C2, (C2×C7⋊C8)⋊18C22, (C2×D4.D7)⋊29C2, (C2×C14).68(C2×D4), (C7×C4○D4)⋊7C22, (C2×C4).97(C22×D7), SmallGroup(448,1228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊6D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 1292 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, Dic28, C2×C7⋊C8, D4⋊D7, D4.D7, D4.D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, C2×C7⋊D4, D4×C14, C7×C4○D4, D28.C4, C8.D14, D8⋊D7, D8⋊3D7, D7×SD16, SD16⋊D7, C2×D4.D7, D4.8D14, C7×C8⋊C22, D4⋊6D14, D4.10D14, D8⋊6D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○SD16, D4×D7, C23×D7, C2×D4×D7, D8⋊6D14
►On 112 points
(1 39 85 21 71 109 46 62)(2 22 47 40 72 63 86 110)(3 41 87 23 73 111 48 64)(4 24 49 42 74 65 88 112)(5 29 89 25 75 99 50 66)(6 26 51 30 76 67 90 100)(7 31 91 27 77 101 52 68)(8 28 53 32 78 69 92 102)(9 33 93 15 79 103 54 70)(10 16 55 34 80 57 94 104)(11 35 95 17 81 105 56 58)(12 18 43 36 82 59 96 106)(13 37 97 19 83 107 44 60)(14 20 45 38 84 61 98 108)
(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 99)(26 100)(27 101)(28 102)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 96)(44 97)(45 98)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)
(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 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 19)(16 18)(20 28)(21 27)(22 26)(23 25)(29 111)(30 110)(31 109)(32 108)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 100)(41 99)(42 112)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 84)(54 83)(55 82)(56 81)(57 59)(60 70)(61 69)(62 68)(63 67)(64 66)
G:=sub<Sym(112)| (1,39,85,21,71,109,46,62)(2,22,47,40,72,63,86,110)(3,41,87,23,73,111,48,64)(4,24,49,42,74,65,88,112)(5,29,89,25,75,99,50,66)(6,26,51,30,76,67,90,100)(7,31,91,27,77,101,52,68)(8,28,53,32,78,69,92,102)(9,33,93,15,79,103,54,70)(10,16,55,34,80,57,94,104)(11,35,95,17,81,105,56,58)(12,18,43,36,82,59,96,106)(13,37,97,19,83,107,44,60)(14,20,45,38,84,61,98,108), (15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,96)(44,97)(45,98)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,19)(16,18)(20,28)(21,27)(22,26)(23,25)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,100)(41,99)(42,112)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,84)(54,83)(55,82)(56,81)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66)>;
G:=Group( (1,39,85,21,71,109,46,62)(2,22,47,40,72,63,86,110)(3,41,87,23,73,111,48,64)(4,24,49,42,74,65,88,112)(5,29,89,25,75,99,50,66)(6,26,51,30,76,67,90,100)(7,31,91,27,77,101,52,68)(8,28,53,32,78,69,92,102)(9,33,93,15,79,103,54,70)(10,16,55,34,80,57,94,104)(11,35,95,17,81,105,56,58)(12,18,43,36,82,59,96,106)(13,37,97,19,83,107,44,60)(14,20,45,38,84,61,98,108), (15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,99)(26,100)(27,101)(28,102)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,96)(44,97)(45,98)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,19)(16,18)(20,28)(21,27)(22,26)(23,25)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,100)(41,99)(42,112)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,84)(54,83)(55,82)(56,81)(57,59)(60,70)(61,69)(62,68)(63,67)(64,66) );
G=PermutationGroup([[(1,39,85,21,71,109,46,62),(2,22,47,40,72,63,86,110),(3,41,87,23,73,111,48,64),(4,24,49,42,74,65,88,112),(5,29,89,25,75,99,50,66),(6,26,51,30,76,67,90,100),(7,31,91,27,77,101,52,68),(8,28,53,32,78,69,92,102),(9,33,93,15,79,103,54,70),(10,16,55,34,80,57,94,104),(11,35,95,17,81,105,56,58),(12,18,43,36,82,59,96,106),(13,37,97,19,83,107,44,60),(14,20,45,38,84,61,98,108)], [(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,99),(26,100),(27,101),(28,102),(29,66),(30,67),(31,68),(32,69),(33,70),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,96),(44,97),(45,98),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,19),(16,18),(20,28),(21,27),(22,26),(23,25),(29,111),(30,110),(31,109),(32,108),(33,107),(34,106),(35,105),(36,104),(37,103),(38,102),(39,101),(40,100),(41,99),(42,112),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,84),(54,83),(55,82),(56,81),(57,59),(60,70),(61,69),(62,68),(63,67),(64,66)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14O | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | D8⋊6D14 |
| kernel | D8⋊6D14 | D28.C4 | C8.D14 | D8⋊D7 | D8⋊3D7 | D7×SD16 | SD16⋊D7 | C2×D4.D7 | D4.8D14 | C7×C8⋊C22 | D4⋊6D14 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
Matrix representation of D8⋊6D14 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 22 |
| 0 | 0 | 36 | 0 | 26 | 0 |
| 0 | 0 | 77 | 13 | 100 | 13 |
| 0 | 0 | 0 | 100 | 100 | 13 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 81 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 0 | 0 | 112 |
| 23 | 10 | 0 | 0 | 0 | 0 |
| 93 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 106 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 103 | 1 | 0 | 0 | 0 | 0 |
| 14 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 112 |
| 0 | 0 | 0 | 112 | 112 | 0 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,77,0,0,0,0,0,13,100,0,0,91,26,100,100,0,0,22,0,13,13],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,81,0,32,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[23,93,0,0,0,0,10,11,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,106,112,1,1,0,0,0,1,0,0],[103,14,0,0,0,0,1,10,0,0,0,0,0,0,112,0,0,0,0,0,7,1,112,112,0,0,0,0,0,112,0,0,0,0,112,0] >;
D8⋊6D14 in GAP, Magma, Sage, TeX
D_8\rtimes_6D_{14} % in TeX
G:=Group("D8:6D14"); // GroupNames label
G:=SmallGroup(448,1228);
// by ID
G=gap.SmallGroup(448,1228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations