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## G = 2+ 1+4.D7order 448 = 26·7

### 1st non-split extension by 2+ 1+4 of 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×Dic14 — D4.9D14 — 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=1, c2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd=fcf-1=a2c, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 564 in 152 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, C22×C14, D4.9D4, C4.Dic7, C4×Dic7, D4.D7, C7⋊Q16, C23.D7, C2×Dic14, D4×C14, D4×C14, C7×C4○D4, C7×C4○D4, C28.D4, D42Dic7, C28.17D4, D4.9D14, C7×2+ 1+4, 2+ 1+4.D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.9D4, C2×C7⋊D4, C24⋊D7, 2+ 1+4.D7

Smallest permutation representation of 2+ 1+4.D7
On 112 points
Generators in S112
```(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)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)
(1 111)(2 112)(3 106)(4 107)(5 108)(6 109)(7 110)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(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)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)
(1 83)(2 84)(3 78)(4 79)(5 80)(6 81)(7 82)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)
(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)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)
(1 33 13 40)(2 32 14 39)(3 31 8 38)(4 30 9 37)(5 29 10 36)(6 35 11 42)(7 34 12 41)(15 45 22 52)(16 44 23 51)(17 43 24 50)(18 49 25 56)(19 48 26 55)(20 47 27 54)(21 46 28 53)(57 108 64 101)(58 107 65 100)(59 106 66 99)(60 112 67 105)(61 111 68 104)(62 110 69 103)(63 109 70 102)(71 87 78 94)(72 86 79 93)(73 85 80 92)(74 91 81 98)(75 90 82 97)(76 89 83 96)(77 88 84 95)```

`G:=sub<Sym(112)| (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)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (1,111)(2,112)(3,106)(4,107)(5,108)(6,109)(7,110)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (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)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (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)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,33,13,40)(2,32,14,39)(3,31,8,38)(4,30,9,37)(5,29,10,36)(6,35,11,42)(7,34,12,41)(15,45,22,52)(16,44,23,51)(17,43,24,50)(18,49,25,56)(19,48,26,55)(20,47,27,54)(21,46,28,53)(57,108,64,101)(58,107,65,100)(59,106,66,99)(60,112,67,105)(61,111,68,104)(62,110,69,103)(63,109,70,102)(71,87,78,94)(72,86,79,93)(73,85,80,92)(74,91,81,98)(75,90,82,97)(76,89,83,96)(77,88,84,95)>;`

`G:=Group( (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)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112), (1,111)(2,112)(3,106)(4,107)(5,108)(6,109)(7,110)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (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)(57,71,64,78)(58,72,65,79)(59,73,66,80)(60,74,67,81)(61,75,68,82)(62,76,69,83)(63,77,70,84)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105), (1,83)(2,84)(3,78)(4,79)(5,80)(6,81)(7,82)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (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)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,33,13,40)(2,32,14,39)(3,31,8,38)(4,30,9,37)(5,29,10,36)(6,35,11,42)(7,34,12,41)(15,45,22,52)(16,44,23,51)(17,43,24,50)(18,49,25,56)(19,48,26,55)(20,47,27,54)(21,46,28,53)(57,108,64,101)(58,107,65,100)(59,106,66,99)(60,112,67,105)(61,111,68,104)(62,110,69,103)(63,109,70,102)(71,87,78,94)(72,86,79,93)(73,85,80,92)(74,91,81,98)(75,90,82,97)(76,89,83,96)(77,88,84,95) );`

`G=PermutationGroup([[(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),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112)], [(1,111),(2,112),(3,106),(4,107),(5,108),(6,109),(7,110),(8,99),(9,100),(10,101),(11,102),(12,103),(13,104),(14,105),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63)], [(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),(57,71,64,78),(58,72,65,79),(59,73,66,80),(60,74,67,81),(61,75,68,82),(62,76,69,83),(63,77,70,84),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105)], [(1,83),(2,84),(3,78),(4,79),(5,80),(6,81),(7,82),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,99),(30,100),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,111),(42,112),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91)], [(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),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,33,13,40),(2,32,14,39),(3,31,8,38),(4,30,9,37),(5,29,10,36),(6,35,11,42),(7,34,12,41),(15,45,22,52),(16,44,23,51),(17,43,24,50),(18,49,25,56),(19,48,26,55),(20,47,27,54),(21,46,28,53),(57,108,64,101),(58,107,65,100),(59,106,66,99),(60,112,67,105),(61,111,68,104),(62,110,69,103),(63,109,70,102),(71,87,78,94),(72,86,79,93),(73,85,80,92),(74,91,81,98),(75,90,82,97),(76,89,83,96),(77,88,84,95)]])`

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14AD 28A ··· 28R order 1 2 2 2 2 2 2 4 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 2 2 4 4 28 28 56 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 D4.9D4 2+ 1+4.D7 kernel 2+ 1+4.D7 C28.D4 D4⋊2Dic7 C28.17D4 D4.9D14 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 15 0 0 0 0 0 0 98 0 0 0 0 0 0 15 0 0 0 0 0 0 98
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 98 0 0 0 0 15 0 0 0 0 98 0 0 0 0 15 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 0 15
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 15 0 0 15 0 0 0 0 0 0 98 0 0
,
 96 47 0 0 0 0 47 96 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
,
 17 47 0 0 0 0 66 96 0 0 0 0 0 0 0 112 0 0 0 0 1 0 0 0 0 0 0 0 0 15 0 0 0 0 15 0

`G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0,0,0,0,0,0,15,0,0,0,0,0,0,98],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,98,0,0,0,0,15,0,0,0,0,98,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,15],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0,0,98,0,0,0,0,0,0,15,0,0],[96,47,0,0,0,0,47,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,66,0,0,0,0,47,96,0,0,0,0,0,0,0,1,0,0,0,0,112,0,0,0,0,0,0,0,0,15,0,0,0,0,15,0] >;`

2+ 1+4.D7 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(448,776);`
`// by ID`

`G=gap.SmallGroup(448,776);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,570,1684,851,438,102,18822]);`
`// Polycyclic`

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

׿
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