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## G = C23.1D14order 224 = 25·7

### 1st non-split extension by C23 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C23.1D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C2×C7⋊D4 — C23.1D14
 Lower central C7 — C14 — C2×C14 — C23.1D14
 Upper central C1 — C2 — C23 — C22⋊C4

Generators and relations for C23.1D14
G = < a,b,c,d | a2=b2=c28=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Smallest permutation representation of C23.1D14
On 56 points
Generators in S56
```(1 29)(3 31)(5 33)(7 35)(9 37)(11 39)(13 41)(15 43)(17 45)(19 47)(21 49)(23 51)(25 53)(27 55)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 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 35 29 7)(2 34)(3 5 31 33)(6 30)(8 28)(9 55 37 27)(10 54)(11 25 39 53)(12 24)(13 51 41 23)(14 50)(15 21 43 49)(16 20)(17 47 45 19)(18 46)(22 42)(26 38)(36 56)(40 52)(44 48)```

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

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

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

C23.1D14 is a maximal subgroup of
C23⋊C45D7  C23⋊D28  C23.5D28  D7×C23⋊C4  (C2×D28)⋊13C4  C24⋊D14  C22⋊C4⋊D14
C23.1D14 is a maximal quotient of
C14.C4≀C2  C4⋊Dic7⋊C4  C23.30D28  (C22×D7)⋊C8  (C2×Dic7)⋊C8  C22.2D56  C7⋊C2≀C4  (C2×C28).D4  C23.D28  C23.2D28  C23.3D28  C23.4D28  (C2×C4).D28  (C2×Q8).D14  C24.2D14

41 conjugacy classes

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

41 irreducible representations

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

Matrix representation of C23.1D14 in GL4(𝔽29) generated by

 28 0 3 3 0 28 12 12 0 0 1 0 0 0 0 1
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 12 14 1 22 19 27 5 12 19 1 9 9 18 18 10 10
,
 13 15 10 22 8 16 22 12 0 0 19 10 0 0 22 10
`G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,3,12,1,0,3,12,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[12,19,19,18,14,27,1,18,1,5,9,10,22,12,9,10],[13,8,0,0,15,16,0,0,10,22,19,22,22,12,10,10] >;`

C23.1D14 in GAP, Magma, Sage, TeX

`C_2^3._1D_{14}`
`% in TeX`

`G:=Group("C2^3.1D14");`
`// GroupNames label`

`G:=SmallGroup(224,12);`
`// by ID`

`G=gap.SmallGroup(224,12);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,297,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^28=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;`
`// generators/relations`

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