metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊5D14, SD16⋊3D14, D28.41D4, D56⋊2C22, C56.2C23, M4(2)⋊9D14, C28.21C24, Dic14.41D4, D28.14C23, Dic14.14C23, (D7×D8)⋊2C2, C7⋊4(D4○D8), C4○D4⋊3D14, C8⋊C22⋊4D7, C7⋊D4.4D4, D8⋊D7⋊3C2, (C2×D4)⋊15D14, C8⋊D14⋊2C2, D56⋊C2⋊2C2, D28.C4⋊1C2, C4.115(D4×D7), (C8×D7)⋊3C22, (C7×D8)⋊3C22, (D4×D7)⋊3C22, C7⋊C8.25C23, D4⋊8D14⋊7C2, D4⋊6D14⋊7C2, C8.2(C22×D7), D4⋊D7⋊14C22, D14.32(C2×D4), C28.242(C2×D4), C4○D28⋊8C22, C56⋊C2⋊3C22, C8⋊D7⋊3C22, Q8⋊D7⋊13C22, C4.21(C23×D7), C22.14(D4×D7), SD16⋊3D7⋊2C2, D4.8D14⋊3C2, (D4×C14)⋊23C22, D4⋊2D7⋊3C22, (C2×D28)⋊36C22, D4.D7⋊13C22, Dic7.37(C2×D4), Q8⋊2D7⋊3C22, (C7×SD16)⋊3C22, (C4×D7).13C23, D4.14(C22×D7), C7⋊Q16⋊12C22, (C7×D4).14C23, Q8.14(C22×D7), (C7×Q8).14C23, (C2×C28).112C23, C14.122(C22×D4), (C7×M4(2))⋊3C22, C2.95(C2×D4×D7), (C2×D4⋊D7)⋊29C2, (C7×C8⋊C22)⋊3C2, (C2×C7⋊C8)⋊17C22, (C2×C14).67(C2×D4), (C7×C4○D4)⋊6C22, (C2×C4).96(C22×D7), SmallGroup(448,1227)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1548 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4, D4 [×2], D4 [×18], Q8, Q8 [×2], C23 [×6], D7 [×5], C14, C14 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8 [×7], SD16 [×2], SD16 [×4], Q16, C2×D4, C2×D4 [×11], C4○D4, C4○D4 [×8], Dic7 [×2], Dic7, C28 [×2], C28, D14 [×2], D14 [×8], C2×C14, C2×C14 [×4], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×5], 2+ (1+4) [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×2], C4×D7 [×2], C4×D7 [×3], D28 [×2], D28 [×2], D28 [×3], C2×Dic7 [×2], C7⋊D4 [×2], C7⋊D4 [×7], C2×C28, C2×C28, C7×D4, C7×D4 [×2], C7×D4 [×2], C7×Q8, C22×D7 [×5], C22×C14, D4○D8, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56 [×2], C2×C7⋊C8, D4⋊D7, D4⋊D7 [×4], D4.D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×D8 [×2], C7×SD16 [×2], C2×D28, C2×D28, C4○D28 [×2], C4○D28, D4×D7 [×4], D4×D7 [×3], D4⋊2D7 [×2], D4⋊2D7, Q8⋊2D7 [×2], C2×C7⋊D4 [×2], D4×C14, C7×C4○D4, D28.C4, C8⋊D14, D7×D8 [×2], D8⋊D7 [×2], D56⋊C2 [×2], SD16⋊3D7 [×2], C2×D4⋊D7, D4.8D14, C7×C8⋊C22, D4⋊6D14, D4⋊8D14, D8⋊5D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D4○D8, D4×D7 [×2], C23×D7, C2×D4×D7, D8⋊5D14
Generators and relations
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >
►On 112 points
(1 25 84 110 62 43 30 90)(2 44 71 91 63 26 31 111)(3 27 72 112 64 45 32 92)(4 46 73 93 65 28 33 99)(5 15 74 100 66 47 34 94)(6 48 75 95 67 16 35 101)(7 17 76 102 68 49 36 96)(8 50 77 97 69 18 37 103)(9 19 78 104 70 51 38 98)(10 52 79 85 57 20 39 105)(11 21 80 106 58 53 40 86)(12 54 81 87 59 22 41 107)(13 23 82 108 60 55 42 88)(14 56 83 89 61 24 29 109)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 68)(30 69)(31 70)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 28)(12 27)(13 26)(14 25)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 112)(42 111)(43 61)(44 60)(45 59)(46 58)(47 57)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(71 88)(72 87)(73 86)(74 85)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)
G:=sub<Sym(112)| (1,25,84,110,62,43,30,90)(2,44,71,91,63,26,31,111)(3,27,72,112,64,45,32,92)(4,46,73,93,65,28,33,99)(5,15,74,100,66,47,34,94)(6,48,75,95,67,16,35,101)(7,17,76,102,68,49,36,96)(8,50,77,97,69,18,37,103)(9,19,78,104,70,51,38,98)(10,52,79,85,57,20,39,105)(11,21,80,106,58,53,40,86)(12,54,81,87,59,22,41,107)(13,23,82,108,60,55,42,88)(14,56,83,89,61,24,29,109), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,68)(30,69)(31,70)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,28)(12,27)(13,26)(14,25)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,112)(42,111)(43,61)(44,60)(45,59)(46,58)(47,57)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)>;
G:=Group( (1,25,84,110,62,43,30,90)(2,44,71,91,63,26,31,111)(3,27,72,112,64,45,32,92)(4,46,73,93,65,28,33,99)(5,15,74,100,66,47,34,94)(6,48,75,95,67,16,35,101)(7,17,76,102,68,49,36,96)(8,50,77,97,69,18,37,103)(9,19,78,104,70,51,38,98)(10,52,79,85,57,20,39,105)(11,21,80,106,58,53,40,86)(12,54,81,87,59,22,41,107)(13,23,82,108,60,55,42,88)(14,56,83,89,61,24,29,109), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,68)(30,69)(31,70)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,28)(12,27)(13,26)(14,25)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,112)(42,111)(43,61)(44,60)(45,59)(46,58)(47,57)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89) );
G=PermutationGroup([(1,25,84,110,62,43,30,90),(2,44,71,91,63,26,31,111),(3,27,72,112,64,45,32,92),(4,46,73,93,65,28,33,99),(5,15,74,100,66,47,34,94),(6,48,75,95,67,16,35,101),(7,17,76,102,68,49,36,96),(8,50,77,97,69,18,37,103),(9,19,78,104,70,51,38,98),(10,52,79,85,57,20,39,105),(11,21,80,106,58,53,40,86),(12,54,81,87,59,22,41,107),(13,23,82,108,60,55,42,88),(14,56,83,89,61,24,29,109)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,54),(16,55),(17,56),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,68),(30,69),(31,70),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,28),(12,27),(13,26),(14,25),(29,110),(30,109),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,100),(40,99),(41,112),(42,111),(43,61),(44,60),(45,59),(46,58),(47,57),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(71,88),(72,87),(73,86),(74,85),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89)])
Matrix representation ►G ⊆ GL8(𝔽113)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 99 | 13 | 8 |
| 0 | 0 | 0 | 0 | 8 | 38 | 16 | 11 |
| 0 | 0 | 0 | 0 | 89 | 59 | 69 | 69 |
| 0 | 0 | 0 | 0 | 54 | 50 | 102 | 51 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 110 | 64 | 0 |
| 0 | 0 | 0 | 0 | 86 | 70 | 0 | 111 |
| 0 | 0 | 0 | 0 | 71 | 97 | 110 | 67 |
| 0 | 0 | 0 | 0 | 88 | 4 | 40 | 43 |
| 104 | 33 | 0 | 0 | 0 | 0 | 0 | 0 |
| 80 | 33 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 104 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 80 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 39 | 64 | 44 |
| 0 | 0 | 0 | 0 | 27 | 43 | 0 | 2 |
| 0 | 0 | 0 | 0 | 81 | 74 | 110 | 51 |
| 0 | 0 | 0 | 0 | 61 | 75 | 40 | 41 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 73 | 60 | 62 | 99 |
| 0 | 0 | 0 | 0 | 72 | 35 | 105 | 51 |
G:=sub<GL(8,GF(113))| [0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,68,8,89,54,0,0,0,0,99,38,59,50,0,0,0,0,13,16,69,102,0,0,0,0,8,11,69,51],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,86,71,88,0,0,0,0,110,70,97,4,0,0,0,0,64,0,110,40,0,0,0,0,0,111,67,43],[104,80,0,0,0,0,0,0,33,33,0,0,0,0,0,0,0,0,104,80,0,0,0,0,0,0,33,33,0,0,0,0,0,0,0,0,32,27,81,61,0,0,0,0,39,43,74,75,0,0,0,0,64,0,110,40,0,0,0,0,44,2,51,41],[0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,0,28,73,72,0,0,0,0,109,0,60,35,0,0,0,0,0,0,62,105,0,0,0,0,0,0,99,51] >;
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 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 | 28 | 28 | 2 | 2 | 4 | 14 | 14 | 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○D8 | D4×D7 | D4×D7 | D8⋊5D14 |
| kernel | D8⋊5D14 | D28.C4 | C8⋊D14 | D7×D8 | D8⋊D7 | D56⋊C2 | SD16⋊3D7 | C2×D4⋊D7 | D4.8D14 | C7×C8⋊C22 | D4⋊6D14 | D4⋊8D14 | 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 |
In GAP, Magma, Sage, TeX
D_8\rtimes_5D_{14} % in TeX
G:=Group("D8:5D14"); // GroupNames label
G:=SmallGroup(448,1227);
// by ID
G=gap.SmallGroup(448,1227);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations