metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊10D14, Q16⋊9D14, SD16⋊10D14, D56⋊22C22, C28.15C24, C56.37C23, D28.10C23, Dic28⋊19C22, Dic14.10C23, C4○D8⋊3D7, C4○D4⋊8D14, (C2×C8)⋊12D14, C7⋊C8.6C23, D8⋊D7⋊7C2, D56⋊C2⋊7C2, (C2×C56)⋊5C22, (C4×D7).51D4, C4.222(D4×D7), (D4×D7)⋊7C22, D56⋊7C2⋊7C2, (Q8×D7)⋊8C22, C22.5(D4×D7), D4⋊D7⋊13C22, Q16⋊D7⋊7C2, D14.53(C2×D4), C28.381(C2×D4), SD16⋊D7⋊7C2, C4○D28⋊6C22, (C7×D8)⋊15C22, Q8⋊D7⋊12C22, D4.9(C22×D7), (C7×D4).9C23, (C4×D7).8C23, C8.15(C22×D7), C4.15(C23×D7), D4.8D14⋊2C2, Q8.9(C22×D7), (C7×Q8).9C23, C8⋊D7⋊15C22, C56⋊C2⋊16C22, C7⋊2(D8⋊C22), D4.D7⋊12C22, (C2×Dic7).80D4, Dic7.58(C2×D4), (C7×Q16)⋊13C22, (C22×D7).42D4, C7⋊Q16⋊11C22, (C2×C28).532C23, (C7×SD16)⋊10C22, D4⋊2D7.5C22, C14.116(C22×D4), Q8⋊2D7.5C22, C2.89(C2×D4×D7), (C7×C4○D8)⋊3C2, (D7×C4○D4)⋊2C2, (C2×C8⋊D7)⋊1C2, (C2×C7⋊C8)⋊16C22, (C2×C14).12(C2×D4), (C7×C4○D4)⋊2C22, (C2×C4×D7).159C22, (C2×C4).619(C22×D7), SmallGroup(448,1221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊10D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=dbd=a4b, dcd=c-1 >
Subgroups: 1332 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C4○D8, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, D8⋊C22, C8⋊D7, C56⋊C2, D56, Dic28, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C7×C4○D4, C2×C8⋊D7, D56⋊7C2, D8⋊D7, D56⋊C2, SD16⋊D7, Q16⋊D7, D4.8D14, C7×C4○D8, D7×C4○D4, D8⋊10D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D8⋊C22, D4×D7, C23×D7, C2×D4×D7, D8⋊10D14
►On 112 points
(1 51 38 16 46 9 33 23)(2 52 39 17 47 10 34 24)(3 53 40 18 48 11 35 25)(4 54 41 19 49 12 29 26)(5 55 42 20 43 13 30 27)(6 56 36 21 44 14 31 28)(7 50 37 15 45 8 32 22)(57 107 83 95 64 100 76 88)(58 108 84 96 65 101 77 89)(59 109 71 97 66 102 78 90)(60 110 72 98 67 103 79 91)(61 111 73 85 68 104 80 92)(62 112 74 86 69 105 81 93)(63 99 75 87 70 106 82 94)
(1 109)(2 103)(3 111)(4 105)(5 99)(6 107)(7 101)(8 58)(9 66)(10 60)(11 68)(12 62)(13 70)(14 64)(15 84)(16 78)(17 72)(18 80)(19 74)(20 82)(21 76)(22 77)(23 71)(24 79)(25 73)(26 81)(27 75)(28 83)(29 93)(30 87)(31 95)(32 89)(33 97)(34 91)(35 85)(36 88)(37 96)(38 90)(39 98)(40 92)(41 86)(42 94)(43 106)(44 100)(45 108)(46 102)(47 110)(48 104)(49 112)(50 65)(51 59)(52 67)(53 61)(54 69)(55 63)(56 57)
(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 44)(2 43)(3 49)(4 48)(5 47)(6 46)(7 45)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 40)(30 39)(31 38)(32 37)(33 36)(34 42)(35 41)(51 56)(52 55)(53 54)(57 66)(58 65)(59 64)(60 63)(61 62)(67 70)(68 69)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 93)(86 92)(87 91)(88 90)(94 98)(95 97)(99 103)(100 102)(104 112)(105 111)(106 110)(107 109)
G:=sub<Sym(112)| (1,51,38,16,46,9,33,23)(2,52,39,17,47,10,34,24)(3,53,40,18,48,11,35,25)(4,54,41,19,49,12,29,26)(5,55,42,20,43,13,30,27)(6,56,36,21,44,14,31,28)(7,50,37,15,45,8,32,22)(57,107,83,95,64,100,76,88)(58,108,84,96,65,101,77,89)(59,109,71,97,66,102,78,90)(60,110,72,98,67,103,79,91)(61,111,73,85,68,104,80,92)(62,112,74,86,69,105,81,93)(63,99,75,87,70,106,82,94), (1,109)(2,103)(3,111)(4,105)(5,99)(6,107)(7,101)(8,58)(9,66)(10,60)(11,68)(12,62)(13,70)(14,64)(15,84)(16,78)(17,72)(18,80)(19,74)(20,82)(21,76)(22,77)(23,71)(24,79)(25,73)(26,81)(27,75)(28,83)(29,93)(30,87)(31,95)(32,89)(33,97)(34,91)(35,85)(36,88)(37,96)(38,90)(39,98)(40,92)(41,86)(42,94)(43,106)(44,100)(45,108)(46,102)(47,110)(48,104)(49,112)(50,65)(51,59)(52,67)(53,61)(54,69)(55,63)(56,57), (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), (1,44)(2,43)(3,49)(4,48)(5,47)(6,46)(7,45)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,40)(30,39)(31,38)(32,37)(33,36)(34,42)(35,41)(51,56)(52,55)(53,54)(57,66)(58,65)(59,64)(60,63)(61,62)(67,70)(68,69)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,103)(100,102)(104,112)(105,111)(106,110)(107,109)>;
G:=Group( (1,51,38,16,46,9,33,23)(2,52,39,17,47,10,34,24)(3,53,40,18,48,11,35,25)(4,54,41,19,49,12,29,26)(5,55,42,20,43,13,30,27)(6,56,36,21,44,14,31,28)(7,50,37,15,45,8,32,22)(57,107,83,95,64,100,76,88)(58,108,84,96,65,101,77,89)(59,109,71,97,66,102,78,90)(60,110,72,98,67,103,79,91)(61,111,73,85,68,104,80,92)(62,112,74,86,69,105,81,93)(63,99,75,87,70,106,82,94), (1,109)(2,103)(3,111)(4,105)(5,99)(6,107)(7,101)(8,58)(9,66)(10,60)(11,68)(12,62)(13,70)(14,64)(15,84)(16,78)(17,72)(18,80)(19,74)(20,82)(21,76)(22,77)(23,71)(24,79)(25,73)(26,81)(27,75)(28,83)(29,93)(30,87)(31,95)(32,89)(33,97)(34,91)(35,85)(36,88)(37,96)(38,90)(39,98)(40,92)(41,86)(42,94)(43,106)(44,100)(45,108)(46,102)(47,110)(48,104)(49,112)(50,65)(51,59)(52,67)(53,61)(54,69)(55,63)(56,57), (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), (1,44)(2,43)(3,49)(4,48)(5,47)(6,46)(7,45)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,40)(30,39)(31,38)(32,37)(33,36)(34,42)(35,41)(51,56)(52,55)(53,54)(57,66)(58,65)(59,64)(60,63)(61,62)(67,70)(68,69)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,103)(100,102)(104,112)(105,111)(106,110)(107,109) );
G=PermutationGroup([[(1,51,38,16,46,9,33,23),(2,52,39,17,47,10,34,24),(3,53,40,18,48,11,35,25),(4,54,41,19,49,12,29,26),(5,55,42,20,43,13,30,27),(6,56,36,21,44,14,31,28),(7,50,37,15,45,8,32,22),(57,107,83,95,64,100,76,88),(58,108,84,96,65,101,77,89),(59,109,71,97,66,102,78,90),(60,110,72,98,67,103,79,91),(61,111,73,85,68,104,80,92),(62,112,74,86,69,105,81,93),(63,99,75,87,70,106,82,94)], [(1,109),(2,103),(3,111),(4,105),(5,99),(6,107),(7,101),(8,58),(9,66),(10,60),(11,68),(12,62),(13,70),(14,64),(15,84),(16,78),(17,72),(18,80),(19,74),(20,82),(21,76),(22,77),(23,71),(24,79),(25,73),(26,81),(27,75),(28,83),(29,93),(30,87),(31,95),(32,89),(33,97),(34,91),(35,85),(36,88),(37,96),(38,90),(39,98),(40,92),(41,86),(42,94),(43,106),(44,100),(45,108),(46,102),(47,110),(48,104),(49,112),(50,65),(51,59),(52,67),(53,61),(54,69),(55,63),(56,57)], [(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)], [(1,44),(2,43),(3,49),(4,48),(5,47),(6,46),(7,45),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,40),(30,39),(31,38),(32,37),(33,36),(34,42),(35,41),(51,56),(52,55),(53,54),(57,66),(58,65),(59,64),(60,63),(61,62),(67,70),(68,69),(71,76),(72,75),(73,74),(77,84),(78,83),(79,82),(80,81),(85,93),(86,92),(87,91),(88,90),(94,98),(95,97),(99,103),(100,102),(104,112),(105,111),(106,110),(107,109)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D8⋊C22 | D4×D7 | D4×D7 | D8⋊10D14 |
| kernel | D8⋊10D14 | C2×C8⋊D7 | D56⋊7C2 | D8⋊D7 | D56⋊C2 | SD16⋊D7 | Q16⋊D7 | D4.8D14 | C7×C4○D8 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8⋊10D14 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 80 | 0 | 0 | 0 | 0 |
| 24 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 89 | 33 | 0 | 0 | 0 | 0 |
| 99 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,98,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1,0,0,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,24,0,0,0,0,80,89,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1],[89,99,0,0,0,0,33,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
D8⋊10D14 in GAP, Magma, Sage, TeX
D_8\rtimes_{10}D_{14} % in TeX
G:=Group("D8:10D14"); // GroupNames label
G:=SmallGroup(448,1221);
// by ID
G=gap.SmallGroup(448,1221);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations