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## G = 2+ 1+4⋊D7order 448 = 26·7

### 1st semidirect product of 2+ 1+4 and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — 2+ 1+4⋊D7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×D28 — D4⋊D14 — 2+ 1+4⋊D7
 Lower central C7 — C14 — C2×C28 — 2+ 1+4⋊D7
 Upper central C1 — C2 — C2×C4 — 2+ 1+4

Generators and relations for 2+ 1+4⋊D7
G = < a,b,c,d,e,f | a4=b2=d2=e7=f2=1, c2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a2c, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 820 in 168 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D7, C14, C14, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C7⋊C8, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, C22×C14, D44D4, C4.Dic7, C4×Dic7, D4⋊D7, Q8⋊D7, C2×D28, C2×C7⋊D4, D4×C14, D4×C14, C7×C4○D4, C7×C4○D4, C28.D4, D42Dic7, C28⋊D4, D4⋊D14, C7×2+ 1+4, 2+ 1+4⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D44D4, C2×C7⋊D4, C24⋊D7, 2+ 1+4⋊D7

Smallest permutation representation of 2+ 1+4⋊D7
On 56 points
Generators in S56
```(1 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 41)(2 42)(3 36)(4 37)(5 38)(6 39)(7 40)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(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 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 55)(2 56)(3 50)(4 51)(5 52)(6 53)(7 54)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 21)(7 20)(8 24)(9 23)(10 22)(11 28)(12 27)(13 26)(14 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14AD 28A ··· 28R order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 14 14 14 14 ··· 14 28 ··· 28 size 1 1 2 4 4 4 4 56 2 2 4 4 28 28 2 2 2 56 56 2 2 2 4 ··· 4 4 ··· 4

67 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 C7⋊D4 C7⋊D4 C7⋊D4 D4⋊4D4 2+ 1+4⋊D7 kernel 2+ 1+4⋊D7 C28.D4 D4⋊2Dic7 C28⋊D4 D4⋊D14 C7×2+ 1+4 C7×D4 C7×Q8 C22×C14 2+ 1+4 C2×D4 C4○D4 D4 Q8 C23 C7 C1 # reps 1 1 2 1 2 1 2 2 2 3 3 6 12 12 12 2 3

Matrix representation of 2+ 1+4⋊D7 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 41 0 0 0 0 22 1 0 0 0 0 0 41 112 2 0 0 1 0 112 1
,
 1 112 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 1 22 91 0 0 112 0 0 0 0 0 112 0 1 112
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 0 0 91 112 0 0 0 0 0 41 112 2 0 0 112 41 112 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 41 112 2 0 0 0 1 22 91 0 0 1 72 0 0 0 0 1 72 1 112
,
 49 47 0 0 0 0 0 30 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 1
,
 83 47 0 0 0 0 53 30 0 0 0 0 0 0 112 41 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 72 1 112

`G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,22,0,1,0,0,41,1,41,0,0,0,0,0,112,112,0,0,0,0,2,1],[1,0,0,0,0,0,112,112,0,0,0,0,0,0,0,0,112,112,0,0,0,1,0,0,0,0,112,22,0,1,0,0,0,91,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,91,0,112,0,0,72,112,41,41,0,0,0,0,112,112,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,41,1,72,72,0,0,112,22,0,1,0,0,2,91,0,112],[49,0,0,0,0,0,47,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[83,53,0,0,0,0,47,30,0,0,0,0,0,0,112,0,0,0,0,0,41,1,0,72,0,0,0,0,1,1,0,0,0,0,0,112] >;`

2+ 1+4⋊D7 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes D_7`
`% in TeX`

`G:=Group("ES+(2,2):D7");`
`// GroupNames label`

`G:=SmallGroup(448,775);`
`// by ID`

`G=gap.SmallGroup(448,775);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,570,1684,851,438,102,18822]);`
`// Polycyclic`

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

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