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G = Q8.D14order 224 = 25·7

5th non-split extension by Q8 of D14 acting via D14/D7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Q8.D14
 Chief series C1 — C7 — C14 — C28 — C4×D7 — Q8⋊2D7 — Q8.D14
 Lower central C7 — C14 — C28 — Q8.D14
 Upper central C1 — C2 — C4 — Q16

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, Q82D7, Q8.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C22×D7, D4×D7, Q8.D14

Smallest permutation representation of 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  SD323D7  Q32⋊D7  Q323D7  D28.30D4  D7×C4○D8  D815D14  D56⋊C22  C56.C23
Q8.D14 is a maximal quotient of
Dic77SD16  Q8.Dic14  Q8⋊Dic7⋊C2  Q82D7⋊C4  D142SD16  D284D4  D14⋊C8.C2  D28.12D4  Dic75D8  C56.4Q8  C8.27(C4×D7)  C87D28  C2.D8⋊D7  D28.2Q8  Q16×Dic7  (C2×Q16)⋊D7  D28.17D4  D143Q16  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`

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