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## G = C28.46D4order 224 = 25·7

### 3rd non-split extension by C28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.46D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C28.46D4
 Lower central C7 — C14 — C2×C14 — C28.46D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C28.46D4
G = < a,b,c | a28=c2=1, b4=a14, bab-1=cac=a-1, cbc=a7b3 >

Smallest permutation representation of C28.46D4
On 56 points
Generators in S56
```(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 49 8 42 15 35 22 56)(2 48 9 41 16 34 23 55)(3 47 10 40 17 33 24 54)(4 46 11 39 18 32 25 53)(5 45 12 38 19 31 26 52)(6 44 13 37 20 30 27 51)(7 43 14 36 21 29 28 50)
(1 8)(2 7)(3 6)(4 5)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(49 56)(50 55)(51 54)(52 53)```

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

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

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

C28.46D4 is a maximal subgroup of
D7×C4.D4  D28.3D4  M4(2).21D14  D28.6D4  D44D28  M4(2)⋊D14  C8.21D28  C8.24D28  M4(2).31D14  D4.3D28  D4.4D28  D2818D4  M4(2).D14  D28.39D4  M4(2).15D14
C28.46D4 is a maximal quotient of
C42.D14  (C22×D7)⋊C8  C4.Dic28  C4.D56  M4(2)⋊Dic7

41 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G 28H 28I 56A ··· 56L order 1 2 2 2 2 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 28 28 2 2 2 2 2 4 4 28 28 2 2 2 4 4 4 2 ··· 2 4 4 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 D4 D7 D14 D28 C7⋊D4 C4×D7 C4.D4 C28.46D4 kernel C28.46D4 C4.Dic7 C7×M4(2) C2×D28 C22×D7 C28 M4(2) C2×C4 C4 C4 C22 C7 C1 # reps 1 1 1 1 4 2 3 3 6 6 6 1 6

Matrix representation of C28.46D4 in GL4(𝔽113) generated by

 58 81 0 0 32 109 0 0 17 78 35 81 74 47 64 19
,
 106 15 107 45 67 62 5 56 15 82 39 4 25 6 93 19
,
 58 75 0 0 32 55 0 0 77 109 94 81 51 10 96 19
`G:=sub<GL(4,GF(113))| [58,32,17,74,81,109,78,47,0,0,35,64,0,0,81,19],[106,67,15,25,15,62,82,6,107,5,39,93,45,56,4,19],[58,32,77,51,75,55,109,10,0,0,94,96,0,0,81,19] >;`

C28.46D4 in GAP, Magma, Sage, TeX

`C_{28}._{46}D_4`
`% in TeX`

`G:=Group("C28.46D4");`
`// GroupNames label`

`G:=SmallGroup(224,29);`
`// by ID`

`G=gap.SmallGroup(224,29);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,86,297,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=c^2=1,b^4=a^14,b*a*b^-1=c*a*c=a^-1,c*b*c=a^7*b^3>;`
`// generators/relations`

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