Copied to
clipboard

## G = D4.D14order 224 = 25·7

### 1st non-split extension by D4 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4.D14
 Chief series C1 — C7 — C14 — C28 — D28 — C4○D28 — D4.D14
 Lower central C7 — C14 — C28 — D4.D14
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for D4.D14
G = < a,b,c,d | a4=b2=1, c14=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c13 >

Subgroups: 254 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C8⋊C22, C7⋊C8, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×C14, C4.Dic7, D4⋊D7, D4.D7, C4○D28, D4×C14, D4.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8⋊C22, C7⋊D4, C22×D7, C2×C7⋊D4, D4.D14

Smallest permutation representation of D4.D14
On 56 points
Generators in S56
```(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(30 44)(32 46)(34 48)(36 50)(38 52)(40 54)(42 56)
(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)
(1 31 15 45)(2 44 16 30)(3 29 17 43)(4 42 18 56)(5 55 19 41)(6 40 20 54)(7 53 21 39)(8 38 22 52)(9 51 23 37)(10 36 24 50)(11 49 25 35)(12 34 26 48)(13 47 27 33)(14 32 28 46)```

`G:=sub<Sym(56)| (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(30,44)(32,46)(34,48)(36,50)(38,52)(40,54)(42,56), (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), (1,31,15,45)(2,44,16,30)(3,29,17,43)(4,42,18,56)(5,55,19,41)(6,40,20,54)(7,53,21,39)(8,38,22,52)(9,51,23,37)(10,36,24,50)(11,49,25,35)(12,34,26,48)(13,47,27,33)(14,32,28,46)>;`

`G:=Group( (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(30,44)(32,46)(34,48)(36,50)(38,52)(40,54)(42,56), (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), (1,31,15,45)(2,44,16,30)(3,29,17,43)(4,42,18,56)(5,55,19,41)(6,40,20,54)(7,53,21,39)(8,38,22,52)(9,51,23,37)(10,36,24,50)(11,49,25,35)(12,34,26,48)(13,47,27,33)(14,32,28,46) );`

`G=PermutationGroup([[(1,22,15,8),(2,23,16,9),(3,24,17,10),(4,25,18,11),(5,26,19,12),(6,27,20,13),(7,28,21,14),(29,36,43,50),(30,37,44,51),(31,38,45,52),(32,39,46,53),(33,40,47,54),(34,41,48,55),(35,42,49,56)], [(1,8),(2,23),(3,10),(4,25),(5,12),(6,27),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(21,28),(30,44),(32,46),(34,48),(36,50),(38,52),(40,54),(42,56)], [(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)], [(1,31,15,45),(2,44,16,30),(3,29,17,43),(4,42,18,56),(5,55,19,41),(6,40,20,54),(7,53,21,39),(8,38,22,52),(9,51,23,37),(10,36,24,50),(11,49,25,35),(12,34,26,48),(13,47,27,33),(14,32,28,46)]])`

41 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 7A 7B 7C 8A 8B 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 4 4 4 7 7 7 8 8 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 4 4 28 2 2 28 2 2 2 28 28 2 ··· 2 4 ··· 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C7⋊D4 C7⋊D4 C8⋊C22 D4.D14 kernel D4.D14 C4.Dic7 D4⋊D7 D4.D7 C4○D28 D4×C14 C28 C2×C14 C2×D4 C2×C4 D4 C4 C22 C7 C1 # reps 1 1 2 2 1 1 1 1 3 3 6 6 6 1 6

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

 1 99 0 0 97 112 0 0 0 0 112 14 0 0 16 1
,
 1 99 0 0 0 112 0 0 0 0 112 0 0 0 16 1
,
 109 56 0 0 64 4 0 0 0 0 28 60 0 0 4 85
,
 0 0 28 60 0 0 4 85 109 56 0 0 64 4 0 0
`G:=sub<GL(4,GF(113))| [1,97,0,0,99,112,0,0,0,0,112,16,0,0,14,1],[1,0,0,0,99,112,0,0,0,0,112,16,0,0,0,1],[109,64,0,0,56,4,0,0,0,0,28,4,0,0,60,85],[0,0,109,64,0,0,56,4,28,4,0,0,60,85,0,0] >;`

D4.D14 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(224,127);`
`// by ID`

`G=gap.SmallGroup(224,127);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,579,159,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^14=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^13>;`
`// generators/relations`

׿
×
𝔽