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## G = D14⋊D4order 224 = 25·7

### 1st semidirect product of D14 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D14⋊D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14⋊D4
 Lower central C7 — C2×C14 — D14⋊D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for D14⋊D4
G = < a,b,c,d | a14=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a12b, dbd=a5b, dcd=c-1 >

Subgroups: 454 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C4⋊D4, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, Dic7⋊C4, D14⋊C4, C7×C22⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D14⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, C4○D28, D4×D7, D14⋊D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 C4○D28 D4×D7 kernel D14⋊D4 Dic7⋊C4 D14⋊C4 C7×C22⋊C4 C2×C4×D7 C2×D28 C2×C7⋊D4 Dic7 D14 C22⋊C4 C14 C2×C4 C23 C2 C2 # reps 1 1 1 1 1 1 2 2 2 3 2 6 3 12 6

Matrix representation of D14⋊D4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 27 21 0 0 0 0 16 20 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 14 27 0 0 0 0 25 15 0 0 0 0 0 0 8 28 0 0 0 0 5 21 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 23 5 0 0 0 0 10 6 0 0 0 0 0 0 2 8 0 0 0 0 25 27 0 0 0 0 0 0 0 1 0 0 0 0 28 0
,
 1 0 0 0 0 0 14 28 0 0 0 0 0 0 2 8 0 0 0 0 25 27 0 0 0 0 0 0 28 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,27,16,0,0,0,0,21,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,25,0,0,0,0,27,15,0,0,0,0,0,0,8,5,0,0,0,0,28,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[23,10,0,0,0,0,5,6,0,0,0,0,0,0,2,25,0,0,0,0,8,27,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,14,0,0,0,0,0,28,0,0,0,0,0,0,2,25,0,0,0,0,8,27,0,0,0,0,0,0,28,0,0,0,0,0,0,1] >;`

D14⋊D4 in GAP, Magma, Sage, TeX

`D_{14}\rtimes D_4`
`% in TeX`

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

`G:=SmallGroup(224,79);`
`// by ID`

`G=gap.SmallGroup(224,79);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,55,506,188,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^14=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^12*b,d*b*d=a^5*b,d*c*d=c^-1>;`
`// generators/relations`

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