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## G = C72⋊2D4order 392 = 23·72

### 1st semidirect product of C72 and D4 acting via D4/C2=C22

Aliases: C722D4, D141D7, C14.3D14, C2.3D72, (D7×C14)⋊1C2, C72(C7⋊D4), C7⋊Dic72C2, (C7×C14).3C22, SmallGroup(392,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7×C14 — C72⋊2D4
 Chief series C1 — C7 — C72 — C7×C14 — D7×C14 — C72⋊2D4
 Lower central C72 — C7×C14 — C72⋊2D4
 Upper central C1 — C2

Generators and relations for C722D4
G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C722D4
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 3 5 7 2 4 6)(8 10 12 14 9 11 13)(15 20 18 16 21 19 17)(22 27 25 23 28 26 24)(29 34 32 30 35 33 31)(36 41 39 37 42 40 38)(43 45 47 49 44 46 48)(50 52 54 56 51 53 55)
(1 51 12 44)(2 50 13 43)(3 56 14 49)(4 55 8 48)(5 54 9 47)(6 53 10 46)(7 52 11 45)(15 41 22 34)(16 40 23 33)(17 39 24 32)(18 38 25 31)(19 37 26 30)(20 36 27 29)(21 42 28 35)
(1 33)(2 34)(3 35)(4 29)(5 30)(6 31)(7 32)(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)```

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 4 7A ··· 7F 7G ··· 7O 14A ··· 14F 14G ··· 14O 14P ··· 14AA order 1 2 2 2 4 7 ··· 7 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 size 1 1 14 14 98 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 14 ··· 14

47 irreducible representations

 dim 1 1 1 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 D4 D7 D14 C7⋊D4 D72 C72⋊2D4 kernel C72⋊2D4 C7⋊Dic7 D7×C14 C72 D14 C14 C7 C2 C1 # reps 1 1 2 1 6 6 12 9 9

Matrix representation of C722D4 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 7 0 0 0 0 22 19 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 15 0 0 0 0 27 7
,
 8 23 0 0 0 0 6 21 0 0 0 0 0 0 28 0 0 0 0 0 22 1 0 0 0 0 0 0 10 18 0 0 0 0 9 19
,
 6 21 0 0 0 0 8 23 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 10 18 0 0 0 0 9 19

`G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,22,0,0,0,0,7,19,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,15,7],[8,6,0,0,0,0,23,21,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,10,9,0,0,0,0,18,19],[6,8,0,0,0,0,21,23,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,9,0,0,0,0,18,19] >;`

C722D4 in GAP, Magma, Sage, TeX

`C_7^2\rtimes_2D_4`
`% in TeX`

`G:=Group("C7^2:2D4");`
`// GroupNames label`

`G:=SmallGroup(392,20);`
`// by ID`

`G=gap.SmallGroup(392,20);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-7,-7,61,488,8404]);`
`// Polycyclic`

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

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