metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D14, C28.50D4, Q8.9D14, C28.19C23, Dic14.12C22, D4.D7⋊6C2, C4○D4.2D7, C7⋊Q16⋊6C2, C7⋊C8.4C22, (C2×C4).23D14, (C2×C14).10D4, C14.61(C2×D4), C7⋊5(C8.C22), C4.25(C7⋊D4), C4.Dic7⋊10C2, (C7×D4).9C22, C4.19(C22×D7), (C2×Dic14)⋊11C2, (C7×Q8).9C22, (C2×C28).44C22, C22.6(C7⋊D4), (C7×C4○D4).3C2, C2.25(C2×C7⋊D4), SmallGroup(224,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.9D14
G = < a,b,c,d | a4=b2=c14=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 198 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C8.C22, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C4.Dic7, D4.D7, C7⋊Q16, C2×Dic14, C7×C4○D4, D4.9D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, C7⋊D4, C22×D7, C2×C7⋊D4, D4.9D14
►On 112 points
Generators in S112
(1 98 108 17)(2 85 109 18)(3 86 110 19)(4 87 111 20)(5 88 112 21)(6 89 99 22)(7 90 100 23)(8 91 101 24)(9 92 102 25)(10 93 103 26)(11 94 104 27)(12 95 105 28)(13 96 106 15)(14 97 107 16)(29 73 47 66)(30 74 48 67)(31 75 49 68)(32 76 50 69)(33 77 51 70)(34 78 52 57)(35 79 53 58)(36 80 54 59)(37 81 55 60)(38 82 56 61)(39 83 43 62)(40 84 44 63)(41 71 45 64)(42 72 46 65)
(1 17)(2 85)(3 19)(4 87)(5 21)(6 89)(7 23)(8 91)(9 25)(10 93)(11 27)(12 95)(13 15)(14 97)(16 107)(18 109)(20 111)(22 99)(24 101)(26 103)(28 105)(29 47)(31 49)(33 51)(35 53)(37 55)(39 43)(41 45)(57 78)(59 80)(61 82)(63 84)(65 72)(67 74)(69 76)(86 110)(88 112)(90 100)(92 102)(94 104)(96 106)(98 108)
(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 66 108 73)(2 65 109 72)(3 64 110 71)(4 63 111 84)(5 62 112 83)(6 61 99 82)(7 60 100 81)(8 59 101 80)(9 58 102 79)(10 57 103 78)(11 70 104 77)(12 69 105 76)(13 68 106 75)(14 67 107 74)(15 31 96 49)(16 30 97 48)(17 29 98 47)(18 42 85 46)(19 41 86 45)(20 40 87 44)(21 39 88 43)(22 38 89 56)(23 37 90 55)(24 36 91 54)(25 35 92 53)(26 34 93 52)(27 33 94 51)(28 32 95 50)
G:=sub<Sym(112)| (1,98,108,17)(2,85,109,18)(3,86,110,19)(4,87,111,20)(5,88,112,21)(6,89,99,22)(7,90,100,23)(8,91,101,24)(9,92,102,25)(10,93,103,26)(11,94,104,27)(12,95,105,28)(13,96,106,15)(14,97,107,16)(29,73,47,66)(30,74,48,67)(31,75,49,68)(32,76,50,69)(33,77,51,70)(34,78,52,57)(35,79,53,58)(36,80,54,59)(37,81,55,60)(38,82,56,61)(39,83,43,62)(40,84,44,63)(41,71,45,64)(42,72,46,65), (1,17)(2,85)(3,19)(4,87)(5,21)(6,89)(7,23)(8,91)(9,25)(10,93)(11,27)(12,95)(13,15)(14,97)(16,107)(18,109)(20,111)(22,99)(24,101)(26,103)(28,105)(29,47)(31,49)(33,51)(35,53)(37,55)(39,43)(41,45)(57,78)(59,80)(61,82)(63,84)(65,72)(67,74)(69,76)(86,110)(88,112)(90,100)(92,102)(94,104)(96,106)(98,108), (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), (1,66,108,73)(2,65,109,72)(3,64,110,71)(4,63,111,84)(5,62,112,83)(6,61,99,82)(7,60,100,81)(8,59,101,80)(9,58,102,79)(10,57,103,78)(11,70,104,77)(12,69,105,76)(13,68,106,75)(14,67,107,74)(15,31,96,49)(16,30,97,48)(17,29,98,47)(18,42,85,46)(19,41,86,45)(20,40,87,44)(21,39,88,43)(22,38,89,56)(23,37,90,55)(24,36,91,54)(25,35,92,53)(26,34,93,52)(27,33,94,51)(28,32,95,50)>;
G:=Group( (1,98,108,17)(2,85,109,18)(3,86,110,19)(4,87,111,20)(5,88,112,21)(6,89,99,22)(7,90,100,23)(8,91,101,24)(9,92,102,25)(10,93,103,26)(11,94,104,27)(12,95,105,28)(13,96,106,15)(14,97,107,16)(29,73,47,66)(30,74,48,67)(31,75,49,68)(32,76,50,69)(33,77,51,70)(34,78,52,57)(35,79,53,58)(36,80,54,59)(37,81,55,60)(38,82,56,61)(39,83,43,62)(40,84,44,63)(41,71,45,64)(42,72,46,65), (1,17)(2,85)(3,19)(4,87)(5,21)(6,89)(7,23)(8,91)(9,25)(10,93)(11,27)(12,95)(13,15)(14,97)(16,107)(18,109)(20,111)(22,99)(24,101)(26,103)(28,105)(29,47)(31,49)(33,51)(35,53)(37,55)(39,43)(41,45)(57,78)(59,80)(61,82)(63,84)(65,72)(67,74)(69,76)(86,110)(88,112)(90,100)(92,102)(94,104)(96,106)(98,108), (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), (1,66,108,73)(2,65,109,72)(3,64,110,71)(4,63,111,84)(5,62,112,83)(6,61,99,82)(7,60,100,81)(8,59,101,80)(9,58,102,79)(10,57,103,78)(11,70,104,77)(12,69,105,76)(13,68,106,75)(14,67,107,74)(15,31,96,49)(16,30,97,48)(17,29,98,47)(18,42,85,46)(19,41,86,45)(20,40,87,44)(21,39,88,43)(22,38,89,56)(23,37,90,55)(24,36,91,54)(25,35,92,53)(26,34,93,52)(27,33,94,51)(28,32,95,50) );
G=PermutationGroup([[(1,98,108,17),(2,85,109,18),(3,86,110,19),(4,87,111,20),(5,88,112,21),(6,89,99,22),(7,90,100,23),(8,91,101,24),(9,92,102,25),(10,93,103,26),(11,94,104,27),(12,95,105,28),(13,96,106,15),(14,97,107,16),(29,73,47,66),(30,74,48,67),(31,75,49,68),(32,76,50,69),(33,77,51,70),(34,78,52,57),(35,79,53,58),(36,80,54,59),(37,81,55,60),(38,82,56,61),(39,83,43,62),(40,84,44,63),(41,71,45,64),(42,72,46,65)], [(1,17),(2,85),(3,19),(4,87),(5,21),(6,89),(7,23),(8,91),(9,25),(10,93),(11,27),(12,95),(13,15),(14,97),(16,107),(18,109),(20,111),(22,99),(24,101),(26,103),(28,105),(29,47),(31,49),(33,51),(35,53),(37,55),(39,43),(41,45),(57,78),(59,80),(61,82),(63,84),(65,72),(67,74),(69,76),(86,110),(88,112),(90,100),(92,102),(94,104),(96,106),(98,108)], [(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)], [(1,66,108,73),(2,65,109,72),(3,64,110,71),(4,63,111,84),(5,62,112,83),(6,61,99,82),(7,60,100,81),(8,59,101,80),(9,58,102,79),(10,57,103,78),(11,70,104,77),(12,69,105,76),(13,68,106,75),(14,67,107,74),(15,31,96,49),(16,30,97,48),(17,29,98,47),(18,42,85,46),(19,41,86,45),(20,40,87,44),(21,39,88,43),(22,38,89,56),(23,37,90,55),(24,36,91,54),(25,35,92,53),(26,34,93,52),(27,33,94,51),(28,32,95,50)]])
D4.9D14 is a maximal subgroup of
M4(2)⋊D14 D4.9D28 D4.3D28 D4.5D28 M4(2).13D14 M4(2).16D14 2+ 1+4.D7 2- 1+4.D7 D8⋊11D14 D8.10D14 SD16⋊D14 D7×C8.C22 C28.C24 D28.33C23 D28.35C23
D4.9D14 is a maximal quotient of
C28.(C2×Q8) C4⋊C4.233D14 (C2×C4).47D28 C28.50D8 C42.51D14 D4.2D28 C28.48SD16 C42.59D14 C28⋊7Q16 (C2×C14).D8 Dic14⋊17D4 C7⋊C8⋊5D4 C22⋊Q8.D7 Dic14.37D4 C7⋊C8.6D4 C42.61D14 C42.62D14 C42.65D14 Dic14.4Q8 C42.68D14 C42.71D14 C4○D4⋊Dic7 (C7×D4).32D4
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | D4.9D14 |
| kernel | D4.9D14 | C4.Dic7 | D4.D7 | C7⋊Q16 | C2×Dic14 | C7×C4○D4 | C28 | C2×C14 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 1 | 6 |
Matrix representation of D4.9D14 ►in GL4(𝔽113) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 100 | 94 |
| 0 | 0 | 19 | 19 |
| 13 | 19 | 0 | 0 |
| 94 | 94 | 0 | 0 |
| 92 | 92 | 92 | 92 |
| 56 | 21 | 56 | 21 |
| 92 | 92 | 21 | 21 |
| 56 | 21 | 57 | 92 |
G:=sub<GL(4,GF(113))| [0,0,112,0,0,0,0,112,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,0,13,94,0,0,19,94,100,19,0,0,94,19,0,0],[92,56,92,56,92,21,92,21,92,56,21,57,92,21,21,92] >;
D4.9D14 in GAP, Magma, Sage, TeX
D_4._9D_{14} % in TeX
G:=Group("D4.9D14"); // GroupNames label
G:=SmallGroup(224,146);
// by ID
G=gap.SmallGroup(224,146);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,188,579,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^14=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations