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## G = 2- 1+4.D7order 448 = 26·7

### The 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=e7=1, c2=d2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd-1=fcf-1=a2c, ce=ec, de=ed, fdf-1=a2cd, fef-1=e-1 >

Subgroups: 500 in 142 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, D4.10D4, C4.Dic7, C4×Dic7, Dic7⋊C4, D4.D7, C7⋊Q16, C2×Dic14, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C28.10D4, D42Dic7, Dic7⋊Q8, 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, D4.10D4, 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 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 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 13 104)(2 112 14 105)(3 106 8 99)(4 107 9 100)(5 108 10 101)(6 109 11 102)(7 110 12 103)(15 85 22 92)(16 86 23 93)(17 87 24 94)(18 88 25 95)(19 89 26 96)(20 90 27 97)(21 91 28 98)(29 78 36 71)(30 79 37 72)(31 80 38 73)(32 81 39 74)(33 82 40 75)(34 83 41 76)(35 84 42 77)(43 57 50 64)(44 58 51 65)(45 59 52 66)(46 60 53 67)(47 61 54 68)(48 62 55 69)(49 63 56 70)
(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 40 13 33)(2 39 14 32)(3 38 8 31)(4 37 9 30)(5 36 10 29)(6 42 11 35)(7 41 12 34)(15 52 22 45)(16 51 23 44)(17 50 24 43)(18 56 25 49)(19 55 26 48)(20 54 27 47)(21 53 28 46)(57 101 64 108)(58 100 65 107)(59 99 66 106)(60 105 67 112)(61 104 68 111)(62 103 69 110)(63 102 70 109)(71 94 78 87)(72 93 79 86)(73 92 80 85)(74 98 81 91)(75 97 82 90)(76 96 83 89)(77 95 84 88)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14R 28A ··· 28AD order 1 2 2 2 2 4 4 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 2 2 4 4 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 D4.10D4 2- 1+4.D7 kernel 2- 1+4.D7 C28.10D4 D4⋊2Dic7 Dic7⋊Q8 D4.9D14 C7×2- 1+4 C2×C28 C7×D4 C7×Q8 2- 1+4 C2×Q8 C4○D4 C2×C4 D4 Q8 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 0 1 0 0 0 0 112 0 0 0 0 0 112 112 1 7 0 0 0 81 32 112
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 112 112 1 7 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 32 81 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 1 1 112 106 0 0 32 0 81 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 50 50 45 102 0 0 18 18 32 100 0 0 95 50 0 0 0 0 102 0 11 45
,
 12 97 0 0 0 0 97 12 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
,
 12 97 0 0 0 0 16 101 0 0 0 0 0 0 63 95 0 0 0 0 95 50 0 0 0 0 63 63 68 11 0 0 0 95 11 45

`G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,112,0,0,0,1,0,112,81,0,0,0,0,1,32,0,0,0,0,7,112],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,112,0,1,32,0,0,1,1,0,81,0,0,7,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,1,32,0,0,1,0,1,0,0,0,0,0,112,81,0,0,0,0,106,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,18,95,102,0,0,50,18,50,0,0,0,45,32,0,11,0,0,102,100,0,45],[12,97,0,0,0,0,97,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,16,0,0,0,0,97,101,0,0,0,0,0,0,63,95,63,0,0,0,95,50,63,95,0,0,0,0,68,11,0,0,0,0,11,45] >;`

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(448,780);`
`// by ID`

`G=gap.SmallGroup(448,780);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,184,570,1684,851,438,102,18822]);`
`// Polycyclic`

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

׿
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