metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊5C2, Q16⋊3D7, D14.3D4, C8.10D14, Q8.5D14, C56.8C22, C28.10C23, Dic7.14D4, D28.5C22, (C8×D7)⋊3C2, C7⋊4(C4○D8), Q8⋊D7⋊4C2, (C7×Q16)⋊3C2, C2.24(D4×D7), C7⋊C8.8C22, Q8⋊2D7⋊3C2, C14.36(C2×D4), C4.10(C22×D7), (C7×Q8).5C22, (C4×D7).12C22, SmallGroup(224,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.D14
G = < a,b,c,d | a4=d2=1, b2=c14=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=a2c13 >
Subgroups: 302 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, D7, C14, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, C28, D14, D14, C4○D8, C7⋊C8, C56, C4×D7, C4×D7, D28, D28, C7×Q8, C8×D7, D56, Q8⋊D7, C7×Q16, Q8⋊2D7, Q8.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C22×D7, D4×D7, Q8.D14
►On 112 points
Generators in S112
(1 59 15 73)(2 74 16 60)(3 61 17 75)(4 76 18 62)(5 63 19 77)(6 78 20 64)(7 65 21 79)(8 80 22 66)(9 67 23 81)(10 82 24 68)(11 69 25 83)(12 84 26 70)(13 71 27 57)(14 58 28 72)(29 103 43 89)(30 90 44 104)(31 105 45 91)(32 92 46 106)(33 107 47 93)(34 94 48 108)(35 109 49 95)(36 96 50 110)(37 111 51 97)(38 98 52 112)(39 85 53 99)(40 100 54 86)(41 87 55 101)(42 102 56 88)
(1 53 15 39)(2 100 16 86)(3 55 17 41)(4 102 18 88)(5 29 19 43)(6 104 20 90)(7 31 21 45)(8 106 22 92)(9 33 23 47)(10 108 24 94)(11 35 25 49)(12 110 26 96)(13 37 27 51)(14 112 28 98)(30 78 44 64)(32 80 46 66)(34 82 48 68)(36 84 50 70)(38 58 52 72)(40 60 54 74)(42 62 56 76)(57 111 71 97)(59 85 73 99)(61 87 75 101)(63 89 77 103)(65 91 79 105)(67 93 81 107)(69 95 83 109)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 56)(26 55)(27 54)(28 53)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)
G:=sub<Sym(112)| (1,59,15,73)(2,74,16,60)(3,61,17,75)(4,76,18,62)(5,63,19,77)(6,78,20,64)(7,65,21,79)(8,80,22,66)(9,67,23,81)(10,82,24,68)(11,69,25,83)(12,84,26,70)(13,71,27,57)(14,58,28,72)(29,103,43,89)(30,90,44,104)(31,105,45,91)(32,92,46,106)(33,107,47,93)(34,94,48,108)(35,109,49,95)(36,96,50,110)(37,111,51,97)(38,98,52,112)(39,85,53,99)(40,100,54,86)(41,87,55,101)(42,102,56,88), (1,53,15,39)(2,100,16,86)(3,55,17,41)(4,102,18,88)(5,29,19,43)(6,104,20,90)(7,31,21,45)(8,106,22,92)(9,33,23,47)(10,108,24,94)(11,35,25,49)(12,110,26,96)(13,37,27,51)(14,112,28,98)(30,78,44,64)(32,80,46,66)(34,82,48,68)(36,84,50,70)(38,58,52,72)(40,60,54,74)(42,62,56,76)(57,111,71,97)(59,85,73,99)(61,87,75,101)(63,89,77,103)(65,91,79,105)(67,93,81,107)(69,95,83,109), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,56)(26,55)(27,54)(28,53)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)>;
G:=Group( (1,59,15,73)(2,74,16,60)(3,61,17,75)(4,76,18,62)(5,63,19,77)(6,78,20,64)(7,65,21,79)(8,80,22,66)(9,67,23,81)(10,82,24,68)(11,69,25,83)(12,84,26,70)(13,71,27,57)(14,58,28,72)(29,103,43,89)(30,90,44,104)(31,105,45,91)(32,92,46,106)(33,107,47,93)(34,94,48,108)(35,109,49,95)(36,96,50,110)(37,111,51,97)(38,98,52,112)(39,85,53,99)(40,100,54,86)(41,87,55,101)(42,102,56,88), (1,53,15,39)(2,100,16,86)(3,55,17,41)(4,102,18,88)(5,29,19,43)(6,104,20,90)(7,31,21,45)(8,106,22,92)(9,33,23,47)(10,108,24,94)(11,35,25,49)(12,110,26,96)(13,37,27,51)(14,112,28,98)(30,78,44,64)(32,80,46,66)(34,82,48,68)(36,84,50,70)(38,58,52,72)(40,60,54,74)(42,62,56,76)(57,111,71,97)(59,85,73,99)(61,87,75,101)(63,89,77,103)(65,91,79,105)(67,93,81,107)(69,95,83,109), (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,56)(26,55)(27,54)(28,53)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101) );
G=PermutationGroup([[(1,59,15,73),(2,74,16,60),(3,61,17,75),(4,76,18,62),(5,63,19,77),(6,78,20,64),(7,65,21,79),(8,80,22,66),(9,67,23,81),(10,82,24,68),(11,69,25,83),(12,84,26,70),(13,71,27,57),(14,58,28,72),(29,103,43,89),(30,90,44,104),(31,105,45,91),(32,92,46,106),(33,107,47,93),(34,94,48,108),(35,109,49,95),(36,96,50,110),(37,111,51,97),(38,98,52,112),(39,85,53,99),(40,100,54,86),(41,87,55,101),(42,102,56,88)], [(1,53,15,39),(2,100,16,86),(3,55,17,41),(4,102,18,88),(5,29,19,43),(6,104,20,90),(7,31,21,45),(8,106,22,92),(9,33,23,47),(10,108,24,94),(11,35,25,49),(12,110,26,96),(13,37,27,51),(14,112,28,98),(30,78,44,64),(32,80,46,66),(34,82,48,68),(36,84,50,70),(38,58,52,72),(40,60,54,74),(42,62,56,76),(57,111,71,97),(59,85,73,99),(61,87,75,101),(63,89,77,103),(65,91,79,105),(67,93,81,107),(69,95,83,109)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,56),(26,55),(27,54),(28,53),(57,100),(58,99),(59,98),(60,97),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,112),(74,111),(75,110),(76,109),(77,108),(78,107),(79,106),(80,105),(81,104),(82,103),(83,102),(84,101)]])
Q8.D14 is a maximal subgroup of
D112⋊C2 SD32⋊3D7 Q32⋊D7 Q32⋊3D7 D28.30D4 D7×C4○D8 D8⋊15D14 D56⋊C22 C56.C23
Q8.D14 is a maximal quotient of
Dic7⋊7SD16 Q8.Dic14 Q8⋊Dic7⋊C2 Q8⋊2D7⋊C4 D14⋊2SD16 D28⋊4D4 D14⋊C8.C2 D28.12D4 Dic7⋊5D8 C56.4Q8 C8.27(C4×D7) C8⋊7D28 C2.D8⋊D7 D28.2Q8 Q16×Dic7 (C2×Q16)⋊D7 D28.17D4 D14⋊3Q16 C56.28D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 28A | 28B | 28C | 28D | ··· | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 14 | 28 | 28 | 2 | 4 | 4 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
35 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 | D4 | D7 | D14 | D14 | C4○D8 | D4×D7 | Q8.D14 |
| kernel | Q8.D14 | C8×D7 | D56 | Q8⋊D7 | C7×Q16 | Q8⋊2D7 | Dic7 | D14 | Q16 | C8 | Q8 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 4 | 3 | 6 |
Matrix representation of Q8.D14 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 112 |
| 0 | 0 | 2 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 15 | 98 |
| 0 | 0 | 0 | 98 |
| 102 | 103 | 0 | 0 |
| 20 | 90 | 0 | 0 |
| 0 | 0 | 26 | 100 |
| 0 | 0 | 26 | 87 |
| 89 | 89 | 0 | 0 |
| 104 | 24 | 0 | 0 |
| 0 | 0 | 0 | 31 |
| 0 | 0 | 62 | 0 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,1,2,0,0,112,112],[112,0,0,0,0,112,0,0,0,0,15,0,0,0,98,98],[102,20,0,0,103,90,0,0,0,0,26,26,0,0,100,87],[89,104,0,0,89,24,0,0,0,0,0,62,0,0,31,0] >;
Q8.D14 in GAP, Magma, Sage, TeX
Q_8.D_{14} % in TeX
G:=Group("Q8.D14"); // GroupNames label
G:=SmallGroup(224,114);
// by ID
G=gap.SmallGroup(224,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,362,116,86,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^14=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=a^2*c^13>;
// generators/relations