metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊1Q8, Dic7.7D4, C4⋊C4⋊4D7, C2.6(Q8×D7), C2.14(D4×D7), C7⋊2(C22⋊Q8), D14⋊C4.2C2, C14.26(C2×D4), (C2×C4).13D14, C14.13(C2×Q8), Dic7⋊C4⋊12C2, (C2×Dic14)⋊4C2, C2.15(C4○D28), C14.13(C4○D4), (C2×C14).37C23, (C2×C28).58C22, C22.51(C22×D7), (C2×Dic7).12C22, (C22×D7).21C22, (C7×C4⋊C4)⋊7C2, (C2×C4×D7).9C2, SmallGroup(224,91)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊Q8
G = < a,b,c,d | a14=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a5b, dbd-1=a12b, dcd-1=c-1 >
Subgroups: 310 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, D7, C14, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic7, Dic7, C28, D14, D14, C2×C14, C22⋊Q8, Dic14, C4×D7, C2×Dic7, C2×C28, C22×D7, Dic7⋊C4, D14⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, D14⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C22×D7, C4○D28, D4×D7, Q8×D7, D14⋊Q8
►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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 72)(16 71)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 87)(30 86)(31 85)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(57 110)(58 109)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 112)(70 111)
(1 36 43 95)(2 35 44 94)(3 34 45 93)(4 33 46 92)(5 32 47 91)(6 31 48 90)(7 30 49 89)(8 29 50 88)(9 42 51 87)(10 41 52 86)(11 40 53 85)(12 39 54 98)(13 38 55 97)(14 37 56 96)(15 108 81 59)(16 107 82 58)(17 106 83 57)(18 105 84 70)(19 104 71 69)(20 103 72 68)(21 102 73 67)(22 101 74 66)(23 100 75 65)(24 99 76 64)(25 112 77 63)(26 111 78 62)(27 110 79 61)(28 109 80 60)
(1 70 43 105)(2 69 44 104)(3 68 45 103)(4 67 46 102)(5 66 47 101)(6 65 48 100)(7 64 49 99)(8 63 50 112)(9 62 51 111)(10 61 52 110)(11 60 53 109)(12 59 54 108)(13 58 55 107)(14 57 56 106)(15 39 81 98)(16 38 82 97)(17 37 83 96)(18 36 84 95)(19 35 71 94)(20 34 72 93)(21 33 73 92)(22 32 74 91)(23 31 75 90)(24 30 76 89)(25 29 77 88)(26 42 78 87)(27 41 79 86)(28 40 80 85)
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,72)(16,71)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,87)(30,86)(31,85)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,112)(70,111), (1,36,43,95)(2,35,44,94)(3,34,45,93)(4,33,46,92)(5,32,47,91)(6,31,48,90)(7,30,49,89)(8,29,50,88)(9,42,51,87)(10,41,52,86)(11,40,53,85)(12,39,54,98)(13,38,55,97)(14,37,56,96)(15,108,81,59)(16,107,82,58)(17,106,83,57)(18,105,84,70)(19,104,71,69)(20,103,72,68)(21,102,73,67)(22,101,74,66)(23,100,75,65)(24,99,76,64)(25,112,77,63)(26,111,78,62)(27,110,79,61)(28,109,80,60), (1,70,43,105)(2,69,44,104)(3,68,45,103)(4,67,46,102)(5,66,47,101)(6,65,48,100)(7,64,49,99)(8,63,50,112)(9,62,51,111)(10,61,52,110)(11,60,53,109)(12,59,54,108)(13,58,55,107)(14,57,56,106)(15,39,81,98)(16,38,82,97)(17,37,83,96)(18,36,84,95)(19,35,71,94)(20,34,72,93)(21,33,73,92)(22,32,74,91)(23,31,75,90)(24,30,76,89)(25,29,77,88)(26,42,78,87)(27,41,79,86)(28,40,80,85)>;
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,72)(16,71)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,87)(30,86)(31,85)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,112)(70,111), (1,36,43,95)(2,35,44,94)(3,34,45,93)(4,33,46,92)(5,32,47,91)(6,31,48,90)(7,30,49,89)(8,29,50,88)(9,42,51,87)(10,41,52,86)(11,40,53,85)(12,39,54,98)(13,38,55,97)(14,37,56,96)(15,108,81,59)(16,107,82,58)(17,106,83,57)(18,105,84,70)(19,104,71,69)(20,103,72,68)(21,102,73,67)(22,101,74,66)(23,100,75,65)(24,99,76,64)(25,112,77,63)(26,111,78,62)(27,110,79,61)(28,109,80,60), (1,70,43,105)(2,69,44,104)(3,68,45,103)(4,67,46,102)(5,66,47,101)(6,65,48,100)(7,64,49,99)(8,63,50,112)(9,62,51,111)(10,61,52,110)(11,60,53,109)(12,59,54,108)(13,58,55,107)(14,57,56,106)(15,39,81,98)(16,38,82,97)(17,37,83,96)(18,36,84,95)(19,35,71,94)(20,34,72,93)(21,33,73,92)(22,32,74,91)(23,31,75,90)(24,30,76,89)(25,29,77,88)(26,42,78,87)(27,41,79,86)(28,40,80,85) );
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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,72),(16,71),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,87),(30,86),(31,85),(32,98),(33,97),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(57,110),(58,109),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,112),(70,111)], [(1,36,43,95),(2,35,44,94),(3,34,45,93),(4,33,46,92),(5,32,47,91),(6,31,48,90),(7,30,49,89),(8,29,50,88),(9,42,51,87),(10,41,52,86),(11,40,53,85),(12,39,54,98),(13,38,55,97),(14,37,56,96),(15,108,81,59),(16,107,82,58),(17,106,83,57),(18,105,84,70),(19,104,71,69),(20,103,72,68),(21,102,73,67),(22,101,74,66),(23,100,75,65),(24,99,76,64),(25,112,77,63),(26,111,78,62),(27,110,79,61),(28,109,80,60)], [(1,70,43,105),(2,69,44,104),(3,68,45,103),(4,67,46,102),(5,66,47,101),(6,65,48,100),(7,64,49,99),(8,63,50,112),(9,62,51,111),(10,61,52,110),(11,60,53,109),(12,59,54,108),(13,58,55,107),(14,57,56,106),(15,39,81,98),(16,38,82,97),(17,37,83,96),(18,36,84,95),(19,35,71,94),(20,34,72,93),(21,33,73,92),(22,32,74,91),(23,31,75,90),(24,30,76,89),(25,29,77,88),(26,42,78,87),(27,41,79,86),(28,40,80,85)]])
D14⋊Q8 is a maximal subgroup of
C14.2- 1+4 C14.102+ 1+4 C14.62- 1+4 C42⋊10D14 C42.93D14 C42.96D14 C42.99D14 C42⋊12D14 Dic14⋊23D4 C42⋊16D14 C42.118D14 C42.122D14 C42.232D14 D28⋊10Q8 C42.133D14 C14.682- 1+4 C14.402+ 1+4 C14.422+ 1+4 C14.492+ 1+4 D7×C22⋊Q8 C14.172- 1+4 Dic14⋊21D4 Dic14⋊22D4 C14.512+ 1+4 C14.522+ 1+4 C14.222- 1+4 C14.582+ 1+4 C14.262- 1+4 C14.792- 1+4 C14.1212+ 1+4 C14.822- 1+4 C4⋊C4⋊28D14 C14.622+ 1+4 C14.832- 1+4 C14.842- 1+4 C14.862- 1+4 C42.236D14 C42.148D14 D28⋊7Q8 C42.150D14 C42.151D14 C42.154D14 C42.157D14 C42.158D14 C42.160D14 C42⋊23D14 C42⋊24D14 C42.189D14 C42.161D14 C42.162D14 C42.164D14 C42.165D14 C42.171D14 D28⋊8Q8 C42.174D14 C42.180D14
D14⋊Q8 is a maximal quotient of
(C2×C28)⋊Q8 Dic7⋊C4⋊C4 C2.(C28⋊Q8) (C2×Dic7).Q8 D14⋊(C4⋊C4) D14⋊C4⋊C4 (C2×C4).20D28 (C22×D7).Q8 Dic14⋊Q8 Dic14.Q8 D28⋊Q8 D28.Q8 Dic14⋊2Q8 Dic14.2Q8 D28⋊2Q8 D28.2Q8 Dic7⋊(C4⋊C4) (C2×C4)⋊Dic14 (C2×C28).287D4 C4⋊C4⋊5Dic7 D14⋊C4⋊6C4 (C2×C28).289D4 (C2×C4).45D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | C4○D4 | D14 | C4○D28 | D4×D7 | Q8×D7 |
| kernel | D14⋊Q8 | Dic7⋊C4 | D14⋊C4 | C7×C4⋊C4 | C2×Dic14 | C2×C4×D7 | Dic7 | D14 | C4⋊C4 | C14 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 9 | 12 | 3 | 3 |
Matrix representation of D14⋊Q8 ►in GL4(𝔽29) generated by
| 6 | 19 | 0 | 0 |
| 20 | 20 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 10 | 1 | 0 | 0 |
| 17 | 19 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 15 | 21 | 0 | 0 |
| 21 | 14 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 28 | 0 |
| 15 | 4 | 0 | 0 |
| 16 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [6,20,0,0,19,20,0,0,0,0,28,0,0,0,0,28],[10,17,0,0,1,19,0,0,0,0,1,0,0,0,0,28],[15,21,0,0,21,14,0,0,0,0,0,28,0,0,28,0],[15,16,0,0,4,14,0,0,0,0,1,0,0,0,0,1] >;
D14⋊Q8 in GAP, Magma, Sage, TeX
D_{14}\rtimes Q_8 % in TeX
G:=Group("D14:Q8"); // GroupNames label
G:=SmallGroup(224,91);
// by ID
G=gap.SmallGroup(224,91);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,55,506,188,86,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^5*b,d*b*d^-1=a^12*b,d*c*d^-1=c^-1>;
// generators/relations