metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14:2Q8, C28.11D4, C4.13D28, C4:C4:5D7, C2.7(Q8xD7), C4:Dic7:6C2, C14.8(C2xD4), C7:3(C22:Q8), D14:C4.3C2, C2.10(C2xD28), (C2xC4).14D14, C14.14(C2xQ8), (C2xDic14):7C2, (C2xC28).6C22, C14.27(C4oD4), (C2xC14).38C23, C2.13(D4:2D7), C22.52(C22xD7), (C2xDic7).13C22, (C22xD7).22C22, (C7xC4:C4):8C2, (C2xC4xD7).3C2, SmallGroup(224,92)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14:2Q8
G = < a,b,c,d | a14=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >
Subgroups: 310 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2xC4, C2xC4, C2xC4, Q8, C23, D7, C14, C22:C4, C4:C4, C4:C4, C22xC4, C2xQ8, Dic7, C28, C28, D14, D14, C2xC14, C22:Q8, Dic14, C4xD7, C2xDic7, C2xDic7, C2xC28, C2xC28, C22xD7, C4:Dic7, D14:C4, C7xC4:C4, C2xDic14, C2xC4xD7, D14:2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2xD4, C2xQ8, C4oD4, D14, C22:Q8, D28, C22xD7, C2xD28, D4:2D7, Q8xD7, D14:2Q8
►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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 56)(13 55)(14 54)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 84)(70 83)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 112)(95 111)(96 110)(97 109)(98 108)
(1 97 54 103)(2 96 55 102)(3 95 56 101)(4 94 43 100)(5 93 44 99)(6 92 45 112)(7 91 46 111)(8 90 47 110)(9 89 48 109)(10 88 49 108)(11 87 50 107)(12 86 51 106)(13 85 52 105)(14 98 53 104)(15 83 42 58)(16 82 29 57)(17 81 30 70)(18 80 31 69)(19 79 32 68)(20 78 33 67)(21 77 34 66)(22 76 35 65)(23 75 36 64)(24 74 37 63)(25 73 38 62)(26 72 39 61)(27 71 40 60)(28 84 41 59)
(1 19 54 32)(2 20 55 33)(3 21 56 34)(4 22 43 35)(5 23 44 36)(6 24 45 37)(7 25 46 38)(8 26 47 39)(9 27 48 40)(10 28 49 41)(11 15 50 42)(12 16 51 29)(13 17 52 30)(14 18 53 31)(57 106 82 86)(58 107 83 87)(59 108 84 88)(60 109 71 89)(61 110 72 90)(62 111 73 91)(63 112 74 92)(64 99 75 93)(65 100 76 94)(66 101 77 95)(67 102 78 96)(68 103 79 97)(69 104 80 98)(70 105 81 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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,56)(13,55)(14,54)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,84)(70,83)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,112)(95,111)(96,110)(97,109)(98,108), (1,97,54,103)(2,96,55,102)(3,95,56,101)(4,94,43,100)(5,93,44,99)(6,92,45,112)(7,91,46,111)(8,90,47,110)(9,89,48,109)(10,88,49,108)(11,87,50,107)(12,86,51,106)(13,85,52,105)(14,98,53,104)(15,83,42,58)(16,82,29,57)(17,81,30,70)(18,80,31,69)(19,79,32,68)(20,78,33,67)(21,77,34,66)(22,76,35,65)(23,75,36,64)(24,74,37,63)(25,73,38,62)(26,72,39,61)(27,71,40,60)(28,84,41,59), (1,19,54,32)(2,20,55,33)(3,21,56,34)(4,22,43,35)(5,23,44,36)(6,24,45,37)(7,25,46,38)(8,26,47,39)(9,27,48,40)(10,28,49,41)(11,15,50,42)(12,16,51,29)(13,17,52,30)(14,18,53,31)(57,106,82,86)(58,107,83,87)(59,108,84,88)(60,109,71,89)(61,110,72,90)(62,111,73,91)(63,112,74,92)(64,99,75,93)(65,100,76,94)(66,101,77,95)(67,102,78,96)(68,103,79,97)(69,104,80,98)(70,105,81,85)>;
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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,56)(13,55)(14,54)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,84)(70,83)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,112)(95,111)(96,110)(97,109)(98,108), (1,97,54,103)(2,96,55,102)(3,95,56,101)(4,94,43,100)(5,93,44,99)(6,92,45,112)(7,91,46,111)(8,90,47,110)(9,89,48,109)(10,88,49,108)(11,87,50,107)(12,86,51,106)(13,85,52,105)(14,98,53,104)(15,83,42,58)(16,82,29,57)(17,81,30,70)(18,80,31,69)(19,79,32,68)(20,78,33,67)(21,77,34,66)(22,76,35,65)(23,75,36,64)(24,74,37,63)(25,73,38,62)(26,72,39,61)(27,71,40,60)(28,84,41,59), (1,19,54,32)(2,20,55,33)(3,21,56,34)(4,22,43,35)(5,23,44,36)(6,24,45,37)(7,25,46,38)(8,26,47,39)(9,27,48,40)(10,28,49,41)(11,15,50,42)(12,16,51,29)(13,17,52,30)(14,18,53,31)(57,106,82,86)(58,107,83,87)(59,108,84,88)(60,109,71,89)(61,110,72,90)(62,111,73,91)(63,112,74,92)(64,99,75,93)(65,100,76,94)(66,101,77,95)(67,102,78,96)(68,103,79,97)(69,104,80,98)(70,105,81,85) );
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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,56),(13,55),(14,54),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,84),(70,83),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,112),(95,111),(96,110),(97,109),(98,108)], [(1,97,54,103),(2,96,55,102),(3,95,56,101),(4,94,43,100),(5,93,44,99),(6,92,45,112),(7,91,46,111),(8,90,47,110),(9,89,48,109),(10,88,49,108),(11,87,50,107),(12,86,51,106),(13,85,52,105),(14,98,53,104),(15,83,42,58),(16,82,29,57),(17,81,30,70),(18,80,31,69),(19,79,32,68),(20,78,33,67),(21,77,34,66),(22,76,35,65),(23,75,36,64),(24,74,37,63),(25,73,38,62),(26,72,39,61),(27,71,40,60),(28,84,41,59)], [(1,19,54,32),(2,20,55,33),(3,21,56,34),(4,22,43,35),(5,23,44,36),(6,24,45,37),(7,25,46,38),(8,26,47,39),(9,27,48,40),(10,28,49,41),(11,15,50,42),(12,16,51,29),(13,17,52,30),(14,18,53,31),(57,106,82,86),(58,107,83,87),(59,108,84,88),(60,109,71,89),(61,110,72,90),(62,111,73,91),(63,112,74,92),(64,99,75,93),(65,100,76,94),(66,101,77,95),(67,102,78,96),(68,103,79,97),(69,104,80,98),(70,105,81,85)]])
D14:2Q8 is a maximal subgroup of
D4.6D28 D14:SD16 C7:C8:1D4 D4.D28 D14:4Q16 Q8.D28 D14:Q16 C7:C8.D4 D14.2SD16 C8:8D28 C28.(C4oD4) C8.2D28 D14.2Q16 C8:3D28 D14:2Q16 C2.D8:7D7 C14.2+ 1+4 C14.102+ 1+4 C14.52- 1+4 C42:8D14 C42.92D14 C42.94D14 C42.98D14 D4:5D28 D4:6D28 C42.229D14 C42.115D14 Q8xD28 Q8:5D28 C42.232D14 C42.134D14 Dic14:19D4 C4:C4:21D14 C14.722- 1+4 D28:20D4 D7xC22:Q8 C14.162- 1+4 D28:22D4 Dic14:21D4 C14.512+ 1+4 C14.1182+ 1+4 C14.242- 1+4 C14.262- 1+4 C14.612+ 1+4 C14.1222+ 1+4 C14.852- 1+4 C14.862- 1+4 C42.148D14 D28:7Q8 C42.237D14 C42.150D14 C42.152D14 C42.155D14 C42.157D14 C42:24D14 C42.161D14 C42.164D14 C42.165D14 C42.241D14 D28:9Q8 C42.177D14 C42.178D14
D14:2Q8 is a maximal quotient of
C4:Dic7:7C4 (C2xDic7):Q8 (C2xC28).28D4 D14:C4:C4 (C2xC4).20D28 (C2xC28).33D4 Dic14.3Q8 D28:3Q8 D28:4Q8 D28.3Q8 C28.7Q16 Dic14:4Q8 (C2xDic7):6Q8 (C2xC4).44D28 C4:(C4:Dic7) C4:(D14:C4) (C2xC4).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 | C4oD4 | D14 | D28 | D4:2D7 | Q8xD7 |
| kernel | D14:2Q8 | C4:Dic7 | D14:C4 | C7xC4:C4 | C2xDic14 | C2xC4xD7 | C28 | D14 | C4:C4 | C14 | C2xC4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 9 | 12 | 3 | 3 |
Matrix representation of D14:2Q8 ►in GL4(F29) generated by
| 4 | 4 | 0 | 0 |
| 25 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 4 | 0 | 0 |
| 18 | 25 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 13 | 5 | 0 | 0 |
| 7 | 16 | 0 | 0 |
| 0 | 0 | 18 | 16 |
| 0 | 0 | 25 | 11 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 13 | 12 |
| 0 | 0 | 10 | 16 |
G:=sub<GL(4,GF(29))| [4,25,0,0,4,18,0,0,0,0,1,0,0,0,0,1],[4,18,0,0,4,25,0,0,0,0,28,0,0,0,0,28],[13,7,0,0,5,16,0,0,0,0,18,25,0,0,16,11],[28,0,0,0,0,28,0,0,0,0,13,10,0,0,12,16] >;
D14:2Q8 in GAP, Magma, Sage, TeX
D_{14}\rtimes_2Q_8 % in TeX
G:=Group("D14:2Q8"); // GroupNames label
G:=SmallGroup(224,92);
// by ID
G=gap.SmallGroup(224,92);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,122,6917]);
// 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=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations