Copied to
clipboard

## G = D14⋊2Q8order 224 = 25·7

### 2nd semidirect product of D14 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D14⋊2Q8
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14⋊2Q8
 Lower central C7 — C2×C14 — D14⋊2Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D142Q8
G = < a,b,c,d | a14=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 310 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, C4⋊C4, C22×C4, C2×Q8, Dic7, C28, C28, D14, D14, C2×C14, C22⋊Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, D142Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, D28, C22×D7, C2×D28, D42D7, Q8×D7, D142Q8

Smallest permutation representation of D142Q8
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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 56)(13 55)(14 54)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 84)(70 83)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 112)(95 111)(96 110)(97 109)(98 108)
(1 97 54 103)(2 96 55 102)(3 95 56 101)(4 94 43 100)(5 93 44 99)(6 92 45 112)(7 91 46 111)(8 90 47 110)(9 89 48 109)(10 88 49 108)(11 87 50 107)(12 86 51 106)(13 85 52 105)(14 98 53 104)(15 83 42 58)(16 82 29 57)(17 81 30 70)(18 80 31 69)(19 79 32 68)(20 78 33 67)(21 77 34 66)(22 76 35 65)(23 75 36 64)(24 74 37 63)(25 73 38 62)(26 72 39 61)(27 71 40 60)(28 84 41 59)
(1 19 54 32)(2 20 55 33)(3 21 56 34)(4 22 43 35)(5 23 44 36)(6 24 45 37)(7 25 46 38)(8 26 47 39)(9 27 48 40)(10 28 49 41)(11 15 50 42)(12 16 51 29)(13 17 52 30)(14 18 53 31)(57 106 82 86)(58 107 83 87)(59 108 84 88)(60 109 71 89)(61 110 72 90)(62 111 73 91)(63 112 74 92)(64 99 75 93)(65 100 76 94)(66 101 77 95)(67 102 78 96)(68 103 79 97)(69 104 80 98)(70 105 81 85)```

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

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

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

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 D28 D4⋊2D7 Q8×D7 kernel D14⋊2Q8 C4⋊Dic7 D14⋊C4 C7×C4⋊C4 C2×Dic14 C2×C4×D7 C28 D14 C4⋊C4 C14 C2×C4 C4 C2 C2 # reps 1 2 2 1 1 1 2 2 3 2 9 12 3 3

Matrix representation of D142Q8 in GL4(𝔽29) generated by

 4 4 0 0 25 18 0 0 0 0 1 0 0 0 0 1
,
 4 4 0 0 18 25 0 0 0 0 28 0 0 0 0 28
,
 13 5 0 0 7 16 0 0 0 0 18 16 0 0 25 11
,
 28 0 0 0 0 28 0 0 0 0 13 12 0 0 10 16
`G:=sub<GL(4,GF(29))| [4,25,0,0,4,18,0,0,0,0,1,0,0,0,0,1],[4,18,0,0,4,25,0,0,0,0,28,0,0,0,0,28],[13,7,0,0,5,16,0,0,0,0,18,25,0,0,16,11],[28,0,0,0,0,28,0,0,0,0,13,10,0,0,12,16] >;`

D142Q8 in GAP, Magma, Sage, TeX

`D_{14}\rtimes_2Q_8`
`% in TeX`

`G:=Group("D14:2Q8");`
`// GroupNames label`

`G:=SmallGroup(224,92);`
`// by ID`

`G=gap.SmallGroup(224,92);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,122,6917]);`
`// Polycyclic`

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

׿
×
𝔽