metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊3Q8, C28.22D4, (C2×Q8)⋊3D7, C2.9(Q8×D7), (Q8×C14)⋊3C2, C7⋊5(C22⋊Q8), D14⋊C4.6C2, C4⋊Dic7⋊15C2, C14.57(C2×D4), (C2×C4).21D14, C14.17(C2×Q8), Dic7⋊C4⋊16C2, C4.18(C7⋊D4), C14.36(C4○D4), (C2×C28).64C22, (C2×C14).58C23, C2.8(Q8⋊2D7), C22.64(C22×D7), (C2×Dic7).21C22, (C22×D7).27C22, (C2×C4×D7).5C2, C2.21(C2×C7⋊D4), SmallGroup(224,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊3Q8
G = < a,b,c,d | a14=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=c-1 >
Subgroups: 286 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, D7, C14, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic7, C28, C28, D14, D14, C2×C14, C22⋊Q8, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C2×C4×D7, Q8×C14, D14⋊3Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C7⋊D4, C22×D7, Q8×D7, Q8⋊2D7, C2×C7⋊D4, D14⋊3Q8
►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 89)(2 88)(3 87)(4 86)(5 85)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 112)(37 111)(38 110)(39 109)(40 108)(41 107)(42 106)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 84)(67 83)(68 82)(69 81)(70 80)
(1 53 90 19)(2 54 91 20)(3 55 92 21)(4 56 93 22)(5 43 94 23)(6 44 95 24)(7 45 96 25)(8 46 97 26)(9 47 98 27)(10 48 85 28)(11 49 86 15)(12 50 87 16)(13 51 88 17)(14 52 89 18)(29 67 103 74)(30 68 104 75)(31 69 105 76)(32 70 106 77)(33 57 107 78)(34 58 108 79)(35 59 109 80)(36 60 110 81)(37 61 111 82)(38 62 112 83)(39 63 99 84)(40 64 100 71)(41 65 101 72)(42 66 102 73)
(1 58 90 79)(2 59 91 80)(3 60 92 81)(4 61 93 82)(5 62 94 83)(6 63 95 84)(7 64 96 71)(8 65 97 72)(9 66 98 73)(10 67 85 74)(11 68 86 75)(12 69 87 76)(13 70 88 77)(14 57 89 78)(15 104 49 30)(16 105 50 31)(17 106 51 32)(18 107 52 33)(19 108 53 34)(20 109 54 35)(21 110 55 36)(22 111 56 37)(23 112 43 38)(24 99 44 39)(25 100 45 40)(26 101 46 41)(27 102 47 42)(28 103 48 29)
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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,84)(67,83)(68,82)(69,81)(70,80), (1,53,90,19)(2,54,91,20)(3,55,92,21)(4,56,93,22)(5,43,94,23)(6,44,95,24)(7,45,96,25)(8,46,97,26)(9,47,98,27)(10,48,85,28)(11,49,86,15)(12,50,87,16)(13,51,88,17)(14,52,89,18)(29,67,103,74)(30,68,104,75)(31,69,105,76)(32,70,106,77)(33,57,107,78)(34,58,108,79)(35,59,109,80)(36,60,110,81)(37,61,111,82)(38,62,112,83)(39,63,99,84)(40,64,100,71)(41,65,101,72)(42,66,102,73), (1,58,90,79)(2,59,91,80)(3,60,92,81)(4,61,93,82)(5,62,94,83)(6,63,95,84)(7,64,96,71)(8,65,97,72)(9,66,98,73)(10,67,85,74)(11,68,86,75)(12,69,87,76)(13,70,88,77)(14,57,89,78)(15,104,49,30)(16,105,50,31)(17,106,51,32)(18,107,52,33)(19,108,53,34)(20,109,54,35)(21,110,55,36)(22,111,56,37)(23,112,43,38)(24,99,44,39)(25,100,45,40)(26,101,46,41)(27,102,47,42)(28,103,48,29)>;
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), (1,89)(2,88)(3,87)(4,86)(5,85)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,112)(37,111)(38,110)(39,109)(40,108)(41,107)(42,106)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,84)(67,83)(68,82)(69,81)(70,80), (1,53,90,19)(2,54,91,20)(3,55,92,21)(4,56,93,22)(5,43,94,23)(6,44,95,24)(7,45,96,25)(8,46,97,26)(9,47,98,27)(10,48,85,28)(11,49,86,15)(12,50,87,16)(13,51,88,17)(14,52,89,18)(29,67,103,74)(30,68,104,75)(31,69,105,76)(32,70,106,77)(33,57,107,78)(34,58,108,79)(35,59,109,80)(36,60,110,81)(37,61,111,82)(38,62,112,83)(39,63,99,84)(40,64,100,71)(41,65,101,72)(42,66,102,73), (1,58,90,79)(2,59,91,80)(3,60,92,81)(4,61,93,82)(5,62,94,83)(6,63,95,84)(7,64,96,71)(8,65,97,72)(9,66,98,73)(10,67,85,74)(11,68,86,75)(12,69,87,76)(13,70,88,77)(14,57,89,78)(15,104,49,30)(16,105,50,31)(17,106,51,32)(18,107,52,33)(19,108,53,34)(20,109,54,35)(21,110,55,36)(22,111,56,37)(23,112,43,38)(24,99,44,39)(25,100,45,40)(26,101,46,41)(27,102,47,42)(28,103,48,29) );
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)], [(1,89),(2,88),(3,87),(4,86),(5,85),(6,98),(7,97),(8,96),(9,95),(10,94),(11,93),(12,92),(13,91),(14,90),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,112),(37,111),(38,110),(39,109),(40,108),(41,107),(42,106),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,84),(67,83),(68,82),(69,81),(70,80)], [(1,53,90,19),(2,54,91,20),(3,55,92,21),(4,56,93,22),(5,43,94,23),(6,44,95,24),(7,45,96,25),(8,46,97,26),(9,47,98,27),(10,48,85,28),(11,49,86,15),(12,50,87,16),(13,51,88,17),(14,52,89,18),(29,67,103,74),(30,68,104,75),(31,69,105,76),(32,70,106,77),(33,57,107,78),(34,58,108,79),(35,59,109,80),(36,60,110,81),(37,61,111,82),(38,62,112,83),(39,63,99,84),(40,64,100,71),(41,65,101,72),(42,66,102,73)], [(1,58,90,79),(2,59,91,80),(3,60,92,81),(4,61,93,82),(5,62,94,83),(6,63,95,84),(7,64,96,71),(8,65,97,72),(9,66,98,73),(10,67,85,74),(11,68,86,75),(12,69,87,76),(13,70,88,77),(14,57,89,78),(15,104,49,30),(16,105,50,31),(17,106,51,32),(18,107,52,33),(19,108,53,34),(20,109,54,35),(21,110,55,36),(22,111,56,37),(23,112,43,38),(24,99,44,39),(25,100,45,40),(26,101,46,41),(27,102,47,42),(28,103,48,29)]])
D14⋊3Q8 is a maximal subgroup of
D14.1SD16 D14⋊2SD16 D14.Q16 C7⋊(C8⋊D4) D14⋊Q16 D14⋊C8.C2 (C2×C8).D14 C7⋊C8.D4 D14⋊6SD16 C56⋊14D4 Dic14.16D4 C56⋊8D4 D14⋊5Q16 D28.17D4 D14⋊3Q16 C56.36D4 C42.232D14 D28⋊10Q8 C42.131D14 C42.132D14 C42.133D14 C42.134D14 C42.135D14 D7×C22⋊Q8 C4⋊C4⋊26D14 C14.162- 1+4 C14.172- 1+4 C14.512+ 1+4 C14.1182+ 1+4 C14.522+ 1+4 C14.532+ 1+4 C14.202- 1+4 C14.212- 1+4 C14.232- 1+4 C14.772- 1+4 C14.572+ 1+4 C14.582+ 1+4 C14.262- 1+4 C42.137D14 D28⋊10D4 Dic14⋊10D4 C42⋊20D14 C42⋊21D14 C42.234D14 C42.144D14 C42.145D14 D28⋊12D4 D28⋊8Q8 C42.241D14 C42.174D14 D28⋊9Q8 C42.176D14 C42.178D14 C42.180D14 Q8×C7⋊D4 C14.442- 1+4 C14.452- 1+4 C14.1042- 1+4 (C2×C28)⋊15D4 C14.1452+ 1+4 C14.1082- 1+4
D14⋊3Q8 is a maximal quotient of
C28⋊(C4⋊C4) C22.23(Q8×D7) (C2×C28).288D4 (C2×C28).54D4 C4⋊(D14⋊C4) D14⋊C4⋊6C4 (C2×C28).289D4 (C2×C4).45D28 Dic14.4Q8 D28.4Q8 D28⋊5Q8 D28⋊6Q8 Dic14⋊5Q8 Dic14⋊6Q8 C14.C22≀C2 (Q8×C14)⋊7C4 (C22×Q8)⋊D7
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 | C7⋊D4 | Q8×D7 | Q8⋊2D7 |
| kernel | D14⋊3Q8 | Dic7⋊C4 | C4⋊Dic7 | D14⋊C4 | C2×C4×D7 | Q8×C14 | C28 | D14 | C2×Q8 | C14 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 2 | 9 | 12 | 3 | 3 |
Matrix representation of D14⋊3Q8 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 25 | 18 |
| 1 | 0 | 0 | 0 |
| 16 | 28 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 18 | 25 |
| 27 | 22 | 0 | 0 |
| 9 | 2 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 27 | 18 |
| 17 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,4,25,0,0,4,18],[1,16,0,0,0,28,0,0,0,0,4,18,0,0,4,25],[27,9,0,0,22,2,0,0,0,0,11,27,0,0,2,18],[17,11,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
D14⋊3Q8 in GAP, Magma, Sage, TeX
D_{14}\rtimes_3Q_8 % in TeX
G:=Group("D14:3Q8"); // GroupNames label
G:=SmallGroup(224,141);
// by ID
G=gap.SmallGroup(224,141);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,86,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^4=1,d^2=c^2,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=c^-1>;
// generators/relations