metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14.4D4, C23.4D14, C2.8(D4×D7), D14⋊C4⋊5C2, C4⋊Dic7⋊4C2, C22⋊C4⋊3D7, (C2×C4).6D14, C14.19(C2×D4), C23.D7⋊4C2, Dic7⋊C4⋊10C2, C14.8(C4○D4), C2.10(C4○D28), C2.8(D4⋊2D7), (C2×C28).52C22, (C2×C14).24C23, C7⋊1(C22.D4), (C2×Dic7).6C22, C22.42(C22×D7), (C22×C14).13C22, (C22×D7).17C22, (C2×C4×D7)⋊10C2, (C7×C22⋊C4)⋊5C2, (C2×C7⋊D4).3C2, SmallGroup(224,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14.D4
G = < a,b,c,d | a14=b2=c4=1, d2=a7, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=a7c-1 >
Subgroups: 326 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C22.D4, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C2×C7⋊D4, D14.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C22×D7, C4○D28, D4×D7, D4⋊2D7, D14.D4
►On 112 points
(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 16)(2 15)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 98)(38 97)(39 96)(40 95)(41 94)(42 93)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)
(1 63 24 100)(2 64 25 101)(3 65 26 102)(4 66 27 103)(5 67 28 104)(6 68 15 105)(7 69 16 106)(8 70 17 107)(9 57 18 108)(10 58 19 109)(11 59 20 110)(12 60 21 111)(13 61 22 112)(14 62 23 99)(29 50 85 71)(30 51 86 72)(31 52 87 73)(32 53 88 74)(33 54 89 75)(34 55 90 76)(35 56 91 77)(36 43 92 78)(37 44 93 79)(38 45 94 80)(39 46 95 81)(40 47 96 82)(41 48 97 83)(42 49 98 84)
(1 54 8 47)(2 55 9 48)(3 56 10 49)(4 43 11 50)(5 44 12 51)(6 45 13 52)(7 46 14 53)(15 80 22 73)(16 81 23 74)(17 82 24 75)(18 83 25 76)(19 84 26 77)(20 71 27 78)(21 72 28 79)(29 59 36 66)(30 60 37 67)(31 61 38 68)(32 62 39 69)(33 63 40 70)(34 64 41 57)(35 65 42 58)(85 110 92 103)(86 111 93 104)(87 112 94 105)(88 99 95 106)(89 100 96 107)(90 101 97 108)(91 102 98 109)
G:=sub<Sym(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), (1,16)(2,15)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106), (1,63,24,100)(2,64,25,101)(3,65,26,102)(4,66,27,103)(5,67,28,104)(6,68,15,105)(7,69,16,106)(8,70,17,107)(9,57,18,108)(10,58,19,109)(11,59,20,110)(12,60,21,111)(13,61,22,112)(14,62,23,99)(29,50,85,71)(30,51,86,72)(31,52,87,73)(32,53,88,74)(33,54,89,75)(34,55,90,76)(35,56,91,77)(36,43,92,78)(37,44,93,79)(38,45,94,80)(39,46,95,81)(40,47,96,82)(41,48,97,83)(42,49,98,84), (1,54,8,47)(2,55,9,48)(3,56,10,49)(4,43,11,50)(5,44,12,51)(6,45,13,52)(7,46,14,53)(15,80,22,73)(16,81,23,74)(17,82,24,75)(18,83,25,76)(19,84,26,77)(20,71,27,78)(21,72,28,79)(29,59,36,66)(30,60,37,67)(31,61,38,68)(32,62,39,69)(33,63,40,70)(34,64,41,57)(35,65,42,58)(85,110,92,103)(86,111,93,104)(87,112,94,105)(88,99,95,106)(89,100,96,107)(90,101,97,108)(91,102,98,109)>;
G:=Group( (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), (1,16)(2,15)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106), (1,63,24,100)(2,64,25,101)(3,65,26,102)(4,66,27,103)(5,67,28,104)(6,68,15,105)(7,69,16,106)(8,70,17,107)(9,57,18,108)(10,58,19,109)(11,59,20,110)(12,60,21,111)(13,61,22,112)(14,62,23,99)(29,50,85,71)(30,51,86,72)(31,52,87,73)(32,53,88,74)(33,54,89,75)(34,55,90,76)(35,56,91,77)(36,43,92,78)(37,44,93,79)(38,45,94,80)(39,46,95,81)(40,47,96,82)(41,48,97,83)(42,49,98,84), (1,54,8,47)(2,55,9,48)(3,56,10,49)(4,43,11,50)(5,44,12,51)(6,45,13,52)(7,46,14,53)(15,80,22,73)(16,81,23,74)(17,82,24,75)(18,83,25,76)(19,84,26,77)(20,71,27,78)(21,72,28,79)(29,59,36,66)(30,60,37,67)(31,61,38,68)(32,62,39,69)(33,63,40,70)(34,64,41,57)(35,65,42,58)(85,110,92,103)(86,111,93,104)(87,112,94,105)(88,99,95,106)(89,100,96,107)(90,101,97,108)(91,102,98,109) );
G=PermutationGroup([[(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)], [(1,16),(2,15),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(29,92),(30,91),(31,90),(32,89),(33,88),(34,87),(35,86),(36,85),(37,98),(38,97),(39,96),(40,95),(41,94),(42,93),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,105),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,112),(65,111),(66,110),(67,109),(68,108),(69,107),(70,106)], [(1,63,24,100),(2,64,25,101),(3,65,26,102),(4,66,27,103),(5,67,28,104),(6,68,15,105),(7,69,16,106),(8,70,17,107),(9,57,18,108),(10,58,19,109),(11,59,20,110),(12,60,21,111),(13,61,22,112),(14,62,23,99),(29,50,85,71),(30,51,86,72),(31,52,87,73),(32,53,88,74),(33,54,89,75),(34,55,90,76),(35,56,91,77),(36,43,92,78),(37,44,93,79),(38,45,94,80),(39,46,95,81),(40,47,96,82),(41,48,97,83),(42,49,98,84)], [(1,54,8,47),(2,55,9,48),(3,56,10,49),(4,43,11,50),(5,44,12,51),(6,45,13,52),(7,46,14,53),(15,80,22,73),(16,81,23,74),(17,82,24,75),(18,83,25,76),(19,84,26,77),(20,71,27,78),(21,72,28,79),(29,59,36,66),(30,60,37,67),(31,61,38,68),(32,62,39,69),(33,63,40,70),(34,64,41,57),(35,65,42,58),(85,110,92,103),(86,111,93,104),(87,112,94,105),(88,99,95,106),(89,100,96,107),(90,101,97,108),(91,102,98,109)]])
D14.D4 is a maximal subgroup of
C24.27D14 C24.30D14 C24.31D14 C42.93D14 C42.94D14 C42.95D14 C42.99D14 C42⋊12D14 D28⋊24D4 C42⋊16D14 C42.229D14 C42.114D14 C42.116D14 C42.118D14 C42.119D14 C24⋊2D14 C24.33D14 C24.35D14 C24.36D14 C14.342+ 1+4 C14.372+ 1+4 C4⋊C4⋊21D14 C14.732- 1+4 D28⋊20D4 C14.442+ 1+4 C14.472+ 1+4 C14.492+ 1+4 C14.162- 1+4 D28⋊22D4 C14.202- 1+4 C14.212- 1+4 C14.242- 1+4 C14.582+ 1+4 C14.262- 1+4 D7×C22.D4 C14.822- 1+4 C14.1222+ 1+4 C14.622+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.842- 1+4 C14.662+ 1+4 C14.852- 1+4 C14.682+ 1+4 C14.862- 1+4 C42.137D14 C42.141D14 D28⋊10D4 C42⋊20D14 C42⋊21D14 C42.234D14 C42.143D14 C42.145D14 C42⋊23D14 C42⋊24D14 C42.189D14 C42.161D14 C42.162D14 C42.165D14
D14.D4 is a maximal quotient of
Dic7⋊C4⋊C4 C4⋊Dic7⋊8C4 C2.(C28⋊Q8) (C2×Dic7).Q8 C22.58(D4×D7) D14⋊C4⋊C4 (C2×C4).21D28 (C22×D7).9D4 D14.D8 D14.SD16 C8⋊Dic7⋊C2 C56⋊1C4⋊C2 D14.1SD16 D14.Q16 D14⋊C8.C2 (C2×C8).D14 C24.4D14 C24.7D14 C24.8D14 C24.9D14 C24.12D14 C24.14D14 C23.16D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 2 | 2 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C4○D28 | D4×D7 | D4⋊2D7 |
| kernel | D14.D4 | Dic7⋊C4 | C4⋊Dic7 | D14⋊C4 | C23.D7 | C7×C22⋊C4 | C2×C4×D7 | C2×C7⋊D4 | D14 | C22⋊C4 | C14 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 3 | 12 | 3 | 3 |
Matrix representation of D14.D4 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 5 | 21 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 21 | 1 |
| 0 | 0 | 24 | 8 |
| 0 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 27 | 24 |
| 0 | 0 | 1 | 2 |
| 17 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,5,0,0,4,21],[1,0,0,0,0,28,0,0,0,0,21,24,0,0,1,8],[0,17,0,0,12,0,0,0,0,0,27,1,0,0,24,2],[17,0,0,0,0,12,0,0,0,0,17,0,0,0,0,17] >;
D14.D4 in GAP, Magma, Sage, TeX
D_{14}.D_4 % in TeX
G:=Group("D14.D4"); // GroupNames label
G:=SmallGroup(224,78);
// by ID
G=gap.SmallGroup(224,78);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,55,218,188,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^7*c^-1>;
// generators/relations