metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊11D14, Q16⋊10D14, D28.46D4, SD16⋊15D14, C28.17C24, C56.43C23, Dic14.46D4, D28.12C23, Dic14.11C23, C4○D8⋊5D7, C4○D4⋊2D14, (C2×C8)⋊14D14, C7⋊D4.2D4, C7⋊C8.8C23, D8⋊D7⋊6C2, D4⋊D7⋊4C22, (D7×SD16)⋊6C2, C4.144(D4×D7), C7⋊3(D4○SD16), Q8⋊D7⋊3C22, D4⋊8D14⋊6C2, D4⋊D14⋊8C2, (Q8×D7)⋊2C22, C22.9(D4×D7), (C2×C56)⋊17C22, Q16⋊D7⋊6C2, D14.30(C2×D4), C28.350(C2×D4), (C7×D8)⋊16C22, (C8×D7)⋊10C22, D4.D7⋊3C22, C7⋊Q16⋊2C22, (D4×D7).2C22, C8.17(C22×D7), C4.17(C23×D7), SD16⋊3D7⋊6C2, D4.9D14⋊7C2, D4⋊2D7⋊2C22, C56⋊C2⋊21C22, C8⋊D7⋊16C22, Dic7.35(C2×D4), (C7×Q16)⋊14C22, (C7×D4).11C23, (C4×D7).10C23, D4.11(C22×D7), D4.10D14⋊5C2, D28.2C4⋊10C2, Q8.11(C22×D7), (C7×Q8).11C23, (C2×C28).534C23, (C7×SD16)⋊16C22, C4○D28.55C22, C14.118(C22×D4), C4.Dic7⋊31C22, Q8⋊2D7.2C22, (C2×Dic14)⋊38C22, (C2×D28).181C22, C2.91(C2×D4×D7), (C7×C4○D8)⋊7C2, (C2×C56⋊C2)⋊27C2, (C2×C14).14(C2×D4), (C7×C4○D4)⋊4C22, (C2×C4).233(C22×D7), SmallGroup(448,1223)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊11D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >
Subgroups: 1364 in 258 conjugacy classes, 99 normal (53 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, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C7×C4○D4, D28.2C4, C2×C56⋊C2, D8⋊D7, D7×SD16, SD16⋊3D7, Q16⋊D7, D4⋊D14, D4.9D14, C7×C4○D8, D4⋊8D14, D4.10D14, D8⋊11D14
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⋊11D14
►On 112 points
(1 20 33 43 65 100 94 76)(2 21 34 44 66 101 95 77)(3 22 35 45 67 102 96 78)(4 23 36 46 68 103 97 79)(5 24 37 47 69 104 98 80)(6 25 38 48 70 105 85 81)(7 26 39 49 57 106 86 82)(8 27 40 50 58 107 87 83)(9 28 41 51 59 108 88 84)(10 15 42 52 60 109 89 71)(11 16 29 53 61 110 90 72)(12 17 30 54 62 111 91 73)(13 18 31 55 63 112 92 74)(14 19 32 56 64 99 93 75)
(1 76)(2 44)(3 78)(4 46)(5 80)(6 48)(7 82)(8 50)(9 84)(10 52)(11 72)(12 54)(13 74)(14 56)(15 42)(16 90)(17 30)(18 92)(19 32)(20 94)(21 34)(22 96)(23 36)(24 98)(25 38)(26 86)(27 40)(28 88)(29 110)(31 112)(33 100)(35 102)(37 104)(39 106)(41 108)(43 65)(45 67)(47 69)(49 57)(51 59)(53 61)(55 63)(58 83)(60 71)(62 73)(64 75)(66 77)(68 79)(70 81)(85 105)(87 107)(89 109)(91 111)(93 99)(95 101)(97 103)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 42)(13 41)(14 40)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 105)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 112)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 98)(68 97)(69 96)(70 95)(71 73)(74 84)(75 83)(76 82)(77 81)(78 80)
G:=sub<Sym(112)| (1,20,33,43,65,100,94,76)(2,21,34,44,66,101,95,77)(3,22,35,45,67,102,96,78)(4,23,36,46,68,103,97,79)(5,24,37,47,69,104,98,80)(6,25,38,48,70,105,85,81)(7,26,39,49,57,106,86,82)(8,27,40,50,58,107,87,83)(9,28,41,51,59,108,88,84)(10,15,42,52,60,109,89,71)(11,16,29,53,61,110,90,72)(12,17,30,54,62,111,91,73)(13,18,31,55,63,112,92,74)(14,19,32,56,64,99,93,75), (1,76)(2,44)(3,78)(4,46)(5,80)(6,48)(7,82)(8,50)(9,84)(10,52)(11,72)(12,54)(13,74)(14,56)(15,42)(16,90)(17,30)(18,92)(19,32)(20,94)(21,34)(22,96)(23,36)(24,98)(25,38)(26,86)(27,40)(28,88)(29,110)(31,112)(33,100)(35,102)(37,104)(39,106)(41,108)(43,65)(45,67)(47,69)(49,57)(51,59)(53,61)(55,63)(58,83)(60,71)(62,73)(64,75)(66,77)(68,79)(70,81)(85,105)(87,107)(89,109)(91,111)(93,99)(95,101)(97,103), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,42)(13,41)(14,40)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,112)(43,49)(44,48)(45,47)(50,56)(51,55)(52,54)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,98)(68,97)(69,96)(70,95)(71,73)(74,84)(75,83)(76,82)(77,81)(78,80)>;
G:=Group( (1,20,33,43,65,100,94,76)(2,21,34,44,66,101,95,77)(3,22,35,45,67,102,96,78)(4,23,36,46,68,103,97,79)(5,24,37,47,69,104,98,80)(6,25,38,48,70,105,85,81)(7,26,39,49,57,106,86,82)(8,27,40,50,58,107,87,83)(9,28,41,51,59,108,88,84)(10,15,42,52,60,109,89,71)(11,16,29,53,61,110,90,72)(12,17,30,54,62,111,91,73)(13,18,31,55,63,112,92,74)(14,19,32,56,64,99,93,75), (1,76)(2,44)(3,78)(4,46)(5,80)(6,48)(7,82)(8,50)(9,84)(10,52)(11,72)(12,54)(13,74)(14,56)(15,42)(16,90)(17,30)(18,92)(19,32)(20,94)(21,34)(22,96)(23,36)(24,98)(25,38)(26,86)(27,40)(28,88)(29,110)(31,112)(33,100)(35,102)(37,104)(39,106)(41,108)(43,65)(45,67)(47,69)(49,57)(51,59)(53,61)(55,63)(58,83)(60,71)(62,73)(64,75)(66,77)(68,79)(70,81)(85,105)(87,107)(89,109)(91,111)(93,99)(95,101)(97,103), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,42)(13,41)(14,40)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,112)(43,49)(44,48)(45,47)(50,56)(51,55)(52,54)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,98)(68,97)(69,96)(70,95)(71,73)(74,84)(75,83)(76,82)(77,81)(78,80) );
G=PermutationGroup([[(1,20,33,43,65,100,94,76),(2,21,34,44,66,101,95,77),(3,22,35,45,67,102,96,78),(4,23,36,46,68,103,97,79),(5,24,37,47,69,104,98,80),(6,25,38,48,70,105,85,81),(7,26,39,49,57,106,86,82),(8,27,40,50,58,107,87,83),(9,28,41,51,59,108,88,84),(10,15,42,52,60,109,89,71),(11,16,29,53,61,110,90,72),(12,17,30,54,62,111,91,73),(13,18,31,55,63,112,92,74),(14,19,32,56,64,99,93,75)], [(1,76),(2,44),(3,78),(4,46),(5,80),(6,48),(7,82),(8,50),(9,84),(10,52),(11,72),(12,54),(13,74),(14,56),(15,42),(16,90),(17,30),(18,92),(19,32),(20,94),(21,34),(22,96),(23,36),(24,98),(25,38),(26,86),(27,40),(28,88),(29,110),(31,112),(33,100),(35,102),(37,104),(39,106),(41,108),(43,65),(45,67),(47,69),(49,57),(51,59),(53,61),(55,63),(58,83),(60,71),(62,73),(64,75),(66,77),(68,79),(70,81),(85,105),(87,107),(89,109),(91,111),(93,99),(95,101),(97,103)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,42),(13,41),(14,40),(15,111),(16,110),(17,109),(18,108),(19,107),(20,106),(21,105),(22,104),(23,103),(24,102),(25,101),(26,100),(27,99),(28,112),(43,49),(44,48),(45,47),(50,56),(51,55),(52,54),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,98),(68,97),(69,96),(70,95),(71,73),(74,84),(75,83),(76,82),(77,81),(78,80)]])
64 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 | ··· | 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 | 7 | 7 | 7 | 8 | 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 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | D8⋊11D14 |
| kernel | D8⋊11D14 | D28.2C4 | C2×C56⋊C2 | D8⋊D7 | D7×SD16 | SD16⋊3D7 | Q16⋊D7 | D4⋊D14 | D4.9D14 | C7×C4○D8 | D4⋊8D14 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8⋊11D14 ►in GL4(𝔽113) generated by
| 76 | 71 | 37 | 42 |
| 42 | 37 | 71 | 76 |
| 76 | 71 | 76 | 71 |
| 42 | 37 | 42 | 37 |
| 76 | 71 | 37 | 42 |
| 42 | 37 | 71 | 76 |
| 37 | 42 | 37 | 42 |
| 71 | 76 | 71 | 76 |
| 0 | 0 | 4 | 109 |
| 0 | 0 | 4 | 81 |
| 109 | 4 | 0 | 0 |
| 109 | 32 | 0 | 0 |
| 0 | 0 | 33 | 33 |
| 0 | 0 | 104 | 80 |
| 33 | 33 | 0 | 0 |
| 104 | 80 | 0 | 0 |
G:=sub<GL(4,GF(113))| [76,42,76,42,71,37,71,37,37,71,76,42,42,76,71,37],[76,42,37,71,71,37,42,76,37,71,37,71,42,76,42,76],[0,0,109,109,0,0,4,32,4,4,0,0,109,81,0,0],[0,0,33,104,0,0,33,80,33,104,0,0,33,80,0,0] >;
D8⋊11D14 in GAP, Magma, Sage, TeX
D_8\rtimes_{11}D_{14} % in TeX
G:=Group("D8:11D14"); // GroupNames label
G:=SmallGroup(448,1223);
// by ID
G=gap.SmallGroup(448,1223);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations