metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊4D14, SD16⋊2D14, C56.1C23, M4(2)⋊8D14, C28.20C24, Dic28⋊1C22, D28.13C23, Dic14.13C23, C8⋊C22⋊6D7, D8⋊D7⋊2C2, (C2×D4)⋊30D14, D8⋊3D7⋊1C2, D4⋊D7⋊6C22, (C4×D7).43D4, C4.190(D4×D7), (C7×D8)⋊2C22, (C8×D7)⋊2C22, (D4×D7)⋊9C22, C7⋊C8.10C23, C8.1(C22×D7), C4○D4.28D14, D14.54(C2×D4), SD16⋊D7⋊1C2, C28.241(C2×D4), C8.D14⋊1C2, C8⋊D7⋊2C22, C56⋊C2⋊2C22, (D7×M4(2))⋊2C2, D4.D7⋊5C22, (Q8×D7)⋊10C22, C7⋊Q16⋊3C22, C4.20(C23×D7), C22.47(D4×D7), SD16⋊3D7⋊1C2, D4.9D14⋊9C2, (D4×C14)⋊22C22, C7⋊3(D8⋊C22), Dic7.60(C2×D4), (C7×SD16)⋊2C22, D4.13(C22×D7), (C4×D7).30C23, (C22×D7).43D4, (C7×D4).13C23, D4.D14⋊10C2, Q8.13(C22×D7), (C7×Q8).13C23, D4⋊2D7⋊10C22, (C2×C28).111C23, (C2×Dic7).195D4, Q8⋊2D7⋊10C22, C4○D28.28C22, C14.121(C22×D4), (C7×M4(2))⋊2C22, C4.Dic7⋊13C22, (C2×Dic14)⋊39C22, C2.94(C2×D4×D7), (D7×C4○D4)⋊4C2, (C7×C8⋊C22)⋊2C2, (C2×C14).66(C2×D4), (C2×D4⋊2D7)⋊26C2, (C2×C4×D7).161C22, (C2×C4).95(C22×D7), (C7×C4○D4).24C22, SmallGroup(448,1226)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1260 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×2], D4 [×11], Q8, Q8 [×5], C23 [×3], D7 [×3], C14, C14 [×4], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×2], SD16 [×2], SD16 [×6], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8 [×2], C4○D4, C4○D4 [×11], Dic7 [×2], Dic7 [×3], C28 [×2], C28, D14 [×2], D14 [×4], C2×C14, C2×C14 [×5], C2×M4(2), C4○D8 [×4], C8⋊C22, C8⋊C22 [×3], C8.C22 [×4], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], Dic14, Dic14 [×2], Dic14 [×2], C4×D7 [×4], C4×D7 [×3], D28, D28, C2×Dic7, C2×Dic7 [×6], C7⋊D4 [×7], C2×C28, C2×C28, C7×D4, C7×D4 [×2], C7×D4 [×2], C7×Q8, C22×D7, C22×D7, C22×C14, D8⋊C22, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], Dic28 [×2], C4.Dic7, D4⋊D7 [×2], D4.D7 [×4], C7⋊Q16 [×2], C7×M4(2), C7×D8 [×2], C7×SD16 [×2], C2×Dic14, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7 [×4], D4⋊2D7 [×3], Q8×D7, Q8⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C7×C4○D4, D7×M4(2), C8.D14, D8⋊D7 [×2], D8⋊3D7 [×2], SD16⋊D7 [×2], SD16⋊3D7 [×2], D4.D14, D4.9D14, C7×C8⋊C22, C2×D4⋊2D7, D7×C4○D4, SD16⋊D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D8⋊C22, D4×D7 [×2], C23×D7, C2×D4×D7, SD16⋊D14
Generators and relations
G = < a,b,c,d | a8=b2=c14=d2=1, bab=dad=a3, cac-1=a-1, cbc-1=a6b, dbd=a2b, dcd=c-1 >
►On 112 points
(1 40 52 111 87 63 21 76)(2 77 22 64 88 112 53 41)(3 42 54 99 89 65 23 78)(4 79 24 66 90 100 55 29)(5 30 56 101 91 67 25 80)(6 81 26 68 92 102 43 31)(7 32 44 103 93 69 27 82)(8 83 28 70 94 104 45 33)(9 34 46 105 95 57 15 84)(10 71 16 58 96 106 47 35)(11 36 48 107 97 59 17 72)(12 73 18 60 98 108 49 37)(13 38 50 109 85 61 19 74)(14 75 20 62 86 110 51 39)
(1 83)(2 57)(3 71)(4 59)(5 73)(6 61)(7 75)(8 63)(9 77)(10 65)(11 79)(12 67)(13 81)(14 69)(15 64)(16 99)(17 66)(18 101)(19 68)(20 103)(21 70)(22 105)(23 58)(24 107)(25 60)(26 109)(27 62)(28 111)(29 48)(30 98)(31 50)(32 86)(33 52)(34 88)(35 54)(36 90)(37 56)(38 92)(39 44)(40 94)(41 46)(42 96)(43 74)(45 76)(47 78)(49 80)(51 82)(53 84)(55 72)(85 102)(87 104)(89 106)(91 108)(93 110)(95 112)(97 100)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 28)(29 59)(30 58)(31 57)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 46)(44 45)(47 56)(48 55)(49 54)(50 53)(51 52)(71 101)(72 100)(73 99)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 88)(86 87)(89 98)(90 97)(91 96)(92 95)(93 94)
G:=sub<Sym(112)| (1,40,52,111,87,63,21,76)(2,77,22,64,88,112,53,41)(3,42,54,99,89,65,23,78)(4,79,24,66,90,100,55,29)(5,30,56,101,91,67,25,80)(6,81,26,68,92,102,43,31)(7,32,44,103,93,69,27,82)(8,83,28,70,94,104,45,33)(9,34,46,105,95,57,15,84)(10,71,16,58,96,106,47,35)(11,36,48,107,97,59,17,72)(12,73,18,60,98,108,49,37)(13,38,50,109,85,61,19,74)(14,75,20,62,86,110,51,39), (1,83)(2,57)(3,71)(4,59)(5,73)(6,61)(7,75)(8,63)(9,77)(10,65)(11,79)(12,67)(13,81)(14,69)(15,64)(16,99)(17,66)(18,101)(19,68)(20,103)(21,70)(22,105)(23,58)(24,107)(25,60)(26,109)(27,62)(28,111)(29,48)(30,98)(31,50)(32,86)(33,52)(34,88)(35,54)(36,90)(37,56)(38,92)(39,44)(40,94)(41,46)(42,96)(43,74)(45,76)(47,78)(49,80)(51,82)(53,84)(55,72)(85,102)(87,104)(89,106)(91,108)(93,110)(95,112)(97,100), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,28)(29,59)(30,58)(31,57)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,46)(44,45)(47,56)(48,55)(49,54)(50,53)(51,52)(71,101)(72,100)(73,99)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,88)(86,87)(89,98)(90,97)(91,96)(92,95)(93,94)>;
G:=Group( (1,40,52,111,87,63,21,76)(2,77,22,64,88,112,53,41)(3,42,54,99,89,65,23,78)(4,79,24,66,90,100,55,29)(5,30,56,101,91,67,25,80)(6,81,26,68,92,102,43,31)(7,32,44,103,93,69,27,82)(8,83,28,70,94,104,45,33)(9,34,46,105,95,57,15,84)(10,71,16,58,96,106,47,35)(11,36,48,107,97,59,17,72)(12,73,18,60,98,108,49,37)(13,38,50,109,85,61,19,74)(14,75,20,62,86,110,51,39), (1,83)(2,57)(3,71)(4,59)(5,73)(6,61)(7,75)(8,63)(9,77)(10,65)(11,79)(12,67)(13,81)(14,69)(15,64)(16,99)(17,66)(18,101)(19,68)(20,103)(21,70)(22,105)(23,58)(24,107)(25,60)(26,109)(27,62)(28,111)(29,48)(30,98)(31,50)(32,86)(33,52)(34,88)(35,54)(36,90)(37,56)(38,92)(39,44)(40,94)(41,46)(42,96)(43,74)(45,76)(47,78)(49,80)(51,82)(53,84)(55,72)(85,102)(87,104)(89,106)(91,108)(93,110)(95,112)(97,100), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,28)(29,59)(30,58)(31,57)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,46)(44,45)(47,56)(48,55)(49,54)(50,53)(51,52)(71,101)(72,100)(73,99)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)(85,88)(86,87)(89,98)(90,97)(91,96)(92,95)(93,94) );
G=PermutationGroup([(1,40,52,111,87,63,21,76),(2,77,22,64,88,112,53,41),(3,42,54,99,89,65,23,78),(4,79,24,66,90,100,55,29),(5,30,56,101,91,67,25,80),(6,81,26,68,92,102,43,31),(7,32,44,103,93,69,27,82),(8,83,28,70,94,104,45,33),(9,34,46,105,95,57,15,84),(10,71,16,58,96,106,47,35),(11,36,48,107,97,59,17,72),(12,73,18,60,98,108,49,37),(13,38,50,109,85,61,19,74),(14,75,20,62,86,110,51,39)], [(1,83),(2,57),(3,71),(4,59),(5,73),(6,61),(7,75),(8,63),(9,77),(10,65),(11,79),(12,67),(13,81),(14,69),(15,64),(16,99),(17,66),(18,101),(19,68),(20,103),(21,70),(22,105),(23,58),(24,107),(25,60),(26,109),(27,62),(28,111),(29,48),(30,98),(31,50),(32,86),(33,52),(34,88),(35,54),(36,90),(37,56),(38,92),(39,44),(40,94),(41,46),(42,96),(43,74),(45,76),(47,78),(49,80),(51,82),(53,84),(55,72),(85,102),(87,104),(89,106),(91,108),(93,110),(95,112),(97,100)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,28),(29,59),(30,58),(31,57),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,60),(43,46),(44,45),(47,56),(48,55),(49,54),(50,53),(51,52),(71,101),(72,100),(73,99),(74,112),(75,111),(76,110),(77,109),(78,108),(79,107),(80,106),(81,105),(82,104),(83,103),(84,102),(85,88),(86,87),(89,98),(90,97),(91,96),(92,95),(93,94)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 38 | 2 | 44 |
| 0 | 0 | 45 | 94 | 15 | 91 |
| 0 | 0 | 80 | 12 | 7 | 41 |
| 0 | 0 | 86 | 103 | 37 | 101 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 111 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 67 | 105 | 50 |
| 0 | 0 | 103 | 28 | 1 | 8 |
| 103 | 103 | 0 | 0 | 0 | 0 |
| 10 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 81 | 0 |
| 0 | 0 | 45 | 94 | 15 | 91 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 31 | 78 | 25 | 19 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 24 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 81 | 0 |
| 0 | 0 | 69 | 19 | 17 | 22 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 35 | 13 | 94 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,24,45,80,86,0,0,38,94,12,103,0,0,2,15,7,37,0,0,44,91,41,101],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,70,103,0,0,111,1,67,28,0,0,0,0,105,1,0,0,0,0,50,8],[103,10,0,0,0,0,103,89,0,0,0,0,0,0,112,45,0,31,0,0,0,94,0,78,0,0,81,15,1,25,0,0,0,91,0,19],[10,24,0,0,0,0,10,103,0,0,0,0,0,0,112,69,0,36,0,0,0,19,0,35,0,0,81,17,1,13,0,0,0,22,0,94] >;
55 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 | ··· | 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 | 4 | 7 | 7 | 7 | 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 | 7 | 7 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 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 | D8⋊C22 | D4×D7 | D4×D7 | SD16⋊D14 |
| kernel | SD16⋊D14 | D7×M4(2) | C8.D14 | D8⋊D7 | D8⋊3D7 | SD16⋊D7 | SD16⋊3D7 | D4.D14 | D4.9D14 | C7×C8⋊C22 | C2×D4⋊2D7 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | 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 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
In GAP, Magma, Sage, TeX
SD_{16}\rtimes D_{14} % in TeX
G:=Group("SD16:D14"); // GroupNames label
G:=SmallGroup(448,1226);
// by ID
G=gap.SmallGroup(448,1226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=d*a*d=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations