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G = D4.9D14order 224 = 25·7

4th non-split extension by D4 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4.9D14
 Chief series C1 — C7 — C14 — C28 — Dic14 — C2×Dic14 — D4.9D14
 Lower central C7 — C14 — C28 — D4.9D14
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4.9D14
G = < a,b,c,d | a4=b2=c14=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 198 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C8.C22, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C4.Dic7, D4.D7, C7⋊Q16, C2×Dic14, C7×C4○D4, D4.9D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8.C22, C7⋊D4, C22×D7, C2×C7⋊D4, D4.9D14

Smallest permutation representation of D4.9D14
On 112 points
Generators in S112
```(1 98 108 17)(2 85 109 18)(3 86 110 19)(4 87 111 20)(5 88 112 21)(6 89 99 22)(7 90 100 23)(8 91 101 24)(9 92 102 25)(10 93 103 26)(11 94 104 27)(12 95 105 28)(13 96 106 15)(14 97 107 16)(29 73 47 66)(30 74 48 67)(31 75 49 68)(32 76 50 69)(33 77 51 70)(34 78 52 57)(35 79 53 58)(36 80 54 59)(37 81 55 60)(38 82 56 61)(39 83 43 62)(40 84 44 63)(41 71 45 64)(42 72 46 65)
(1 17)(2 85)(3 19)(4 87)(5 21)(6 89)(7 23)(8 91)(9 25)(10 93)(11 27)(12 95)(13 15)(14 97)(16 107)(18 109)(20 111)(22 99)(24 101)(26 103)(28 105)(29 47)(31 49)(33 51)(35 53)(37 55)(39 43)(41 45)(57 78)(59 80)(61 82)(63 84)(65 72)(67 74)(69 76)(86 110)(88 112)(90 100)(92 102)(94 104)(96 106)(98 108)
(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 66 108 73)(2 65 109 72)(3 64 110 71)(4 63 111 84)(5 62 112 83)(6 61 99 82)(7 60 100 81)(8 59 101 80)(9 58 102 79)(10 57 103 78)(11 70 104 77)(12 69 105 76)(13 68 106 75)(14 67 107 74)(15 31 96 49)(16 30 97 48)(17 29 98 47)(18 42 85 46)(19 41 86 45)(20 40 87 44)(21 39 88 43)(22 38 89 56)(23 37 90 55)(24 36 91 54)(25 35 92 53)(26 34 93 52)(27 33 94 51)(28 32 95 50)```

`G:=sub<Sym(112)| (1,98,108,17)(2,85,109,18)(3,86,110,19)(4,87,111,20)(5,88,112,21)(6,89,99,22)(7,90,100,23)(8,91,101,24)(9,92,102,25)(10,93,103,26)(11,94,104,27)(12,95,105,28)(13,96,106,15)(14,97,107,16)(29,73,47,66)(30,74,48,67)(31,75,49,68)(32,76,50,69)(33,77,51,70)(34,78,52,57)(35,79,53,58)(36,80,54,59)(37,81,55,60)(38,82,56,61)(39,83,43,62)(40,84,44,63)(41,71,45,64)(42,72,46,65), (1,17)(2,85)(3,19)(4,87)(5,21)(6,89)(7,23)(8,91)(9,25)(10,93)(11,27)(12,95)(13,15)(14,97)(16,107)(18,109)(20,111)(22,99)(24,101)(26,103)(28,105)(29,47)(31,49)(33,51)(35,53)(37,55)(39,43)(41,45)(57,78)(59,80)(61,82)(63,84)(65,72)(67,74)(69,76)(86,110)(88,112)(90,100)(92,102)(94,104)(96,106)(98,108), (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), (1,66,108,73)(2,65,109,72)(3,64,110,71)(4,63,111,84)(5,62,112,83)(6,61,99,82)(7,60,100,81)(8,59,101,80)(9,58,102,79)(10,57,103,78)(11,70,104,77)(12,69,105,76)(13,68,106,75)(14,67,107,74)(15,31,96,49)(16,30,97,48)(17,29,98,47)(18,42,85,46)(19,41,86,45)(20,40,87,44)(21,39,88,43)(22,38,89,56)(23,37,90,55)(24,36,91,54)(25,35,92,53)(26,34,93,52)(27,33,94,51)(28,32,95,50)>;`

`G:=Group( (1,98,108,17)(2,85,109,18)(3,86,110,19)(4,87,111,20)(5,88,112,21)(6,89,99,22)(7,90,100,23)(8,91,101,24)(9,92,102,25)(10,93,103,26)(11,94,104,27)(12,95,105,28)(13,96,106,15)(14,97,107,16)(29,73,47,66)(30,74,48,67)(31,75,49,68)(32,76,50,69)(33,77,51,70)(34,78,52,57)(35,79,53,58)(36,80,54,59)(37,81,55,60)(38,82,56,61)(39,83,43,62)(40,84,44,63)(41,71,45,64)(42,72,46,65), (1,17)(2,85)(3,19)(4,87)(5,21)(6,89)(7,23)(8,91)(9,25)(10,93)(11,27)(12,95)(13,15)(14,97)(16,107)(18,109)(20,111)(22,99)(24,101)(26,103)(28,105)(29,47)(31,49)(33,51)(35,53)(37,55)(39,43)(41,45)(57,78)(59,80)(61,82)(63,84)(65,72)(67,74)(69,76)(86,110)(88,112)(90,100)(92,102)(94,104)(96,106)(98,108), (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), (1,66,108,73)(2,65,109,72)(3,64,110,71)(4,63,111,84)(5,62,112,83)(6,61,99,82)(7,60,100,81)(8,59,101,80)(9,58,102,79)(10,57,103,78)(11,70,104,77)(12,69,105,76)(13,68,106,75)(14,67,107,74)(15,31,96,49)(16,30,97,48)(17,29,98,47)(18,42,85,46)(19,41,86,45)(20,40,87,44)(21,39,88,43)(22,38,89,56)(23,37,90,55)(24,36,91,54)(25,35,92,53)(26,34,93,52)(27,33,94,51)(28,32,95,50) );`

`G=PermutationGroup([[(1,98,108,17),(2,85,109,18),(3,86,110,19),(4,87,111,20),(5,88,112,21),(6,89,99,22),(7,90,100,23),(8,91,101,24),(9,92,102,25),(10,93,103,26),(11,94,104,27),(12,95,105,28),(13,96,106,15),(14,97,107,16),(29,73,47,66),(30,74,48,67),(31,75,49,68),(32,76,50,69),(33,77,51,70),(34,78,52,57),(35,79,53,58),(36,80,54,59),(37,81,55,60),(38,82,56,61),(39,83,43,62),(40,84,44,63),(41,71,45,64),(42,72,46,65)], [(1,17),(2,85),(3,19),(4,87),(5,21),(6,89),(7,23),(8,91),(9,25),(10,93),(11,27),(12,95),(13,15),(14,97),(16,107),(18,109),(20,111),(22,99),(24,101),(26,103),(28,105),(29,47),(31,49),(33,51),(35,53),(37,55),(39,43),(41,45),(57,78),(59,80),(61,82),(63,84),(65,72),(67,74),(69,76),(86,110),(88,112),(90,100),(92,102),(94,104),(96,106),(98,108)], [(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)], [(1,66,108,73),(2,65,109,72),(3,64,110,71),(4,63,111,84),(5,62,112,83),(6,61,99,82),(7,60,100,81),(8,59,101,80),(9,58,102,79),(10,57,103,78),(11,70,104,77),(12,69,105,76),(13,68,106,75),(14,67,107,74),(15,31,96,49),(16,30,97,48),(17,29,98,47),(18,42,85,46),(19,41,86,45),(20,40,87,44),(21,39,88,43),(22,38,89,56),(23,37,90,55),(24,36,91,54),(25,35,92,53),(26,34,93,52),(27,33,94,51),(28,32,95,50)]])`

41 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14L 28A ··· 28F 28G ··· 28O order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 14 14 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 4 2 2 4 28 28 2 2 2 28 28 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 C7⋊D4 C7⋊D4 C8.C22 D4.9D14 kernel D4.9D14 C4.Dic7 D4.D7 C7⋊Q16 C2×Dic14 C7×C4○D4 C28 C2×C14 C4○D4 C2×C4 D4 Q8 C4 C22 C7 C1 # reps 1 1 2 2 1 1 1 1 3 3 3 3 6 6 1 6

Matrix representation of D4.9D14 in GL4(𝔽113) generated by

 0 0 1 0 0 0 0 1 112 0 0 0 0 112 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 0 0 100 94 0 0 19 19 13 19 0 0 94 94 0 0
,
 92 92 92 92 56 21 56 21 92 92 21 21 56 21 57 92
`G:=sub<GL(4,GF(113))| [0,0,112,0,0,0,0,112,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,0,13,94,0,0,19,94,100,19,0,0,94,19,0,0],[92,56,92,56,92,21,92,21,92,56,21,57,92,21,21,92] >;`

D4.9D14 in GAP, Magma, Sage, TeX

`D_4._9D_{14}`
`% in TeX`

`G:=Group("D4.9D14");`
`// GroupNames label`

`G:=SmallGroup(224,146);`
`// by ID`

`G=gap.SmallGroup(224,146);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,188,579,159,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^14=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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