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

Subgroups: 628 in 146 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, C23, D7, C14, C14, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C7⋊C8, D28, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, D4.8D4, C4.Dic7, C4×Dic7, D14⋊C4, D4⋊D7, Q8⋊D7, C2×D28, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C28.10D4, D42Dic7, C28.23D4, 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, D4.8D4, 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 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 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(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)
(1 104 13 111)(2 105 14 112)(3 99 8 106)(4 100 9 107)(5 101 10 108)(6 102 11 109)(7 103 12 110)(15 92 22 85)(16 93 23 86)(17 94 24 87)(18 95 25 88)(19 96 26 89)(20 97 27 90)(21 98 28 91)(29 71 36 78)(30 72 37 79)(31 73 38 80)(32 74 39 81)(33 75 40 82)(34 76 41 83)(35 77 42 84)(43 64 50 57)(44 65 51 58)(45 66 52 59)(46 67 53 60)(47 68 54 61)(48 69 55 62)(49 70 56 63)
(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)(2 32)(3 31)(4 30)(5 29)(6 35)(7 34)(8 38)(9 37)(10 36)(11 42)(12 41)(13 40)(14 39)(15 45)(16 44)(17 43)(18 49)(19 48)(20 47)(21 46)(22 52)(23 51)(24 50)(25 56)(26 55)(27 54)(28 53)(57 108)(58 107)(59 106)(60 112)(61 111)(62 110)(63 109)(64 101)(65 100)(66 99)(67 105)(68 104)(69 103)(70 102)(71 87)(72 86)(73 85)(74 91)(75 90)(76 89)(77 88)(78 94)(79 93)(80 92)(81 98)(82 97)(83 96)(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)(57,71,64,78)(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), (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(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), (1,104,13,111)(2,105,14,112)(3,99,8,106)(4,100,9,107)(5,101,10,108)(6,102,11,109)(7,103,12,110)(15,92,22,85)(16,93,23,86)(17,94,24,87)(18,95,25,88)(19,96,26,89)(20,97,27,90)(21,98,28,91)(29,71,36,78)(30,72,37,79)(31,73,38,80)(32,74,39,81)(33,75,40,82)(34,76,41,83)(35,77,42,84)(43,64,50,57)(44,65,51,58)(45,66,52,59)(46,67,53,60)(47,68,54,61)(48,69,55,62)(49,70,56,63), (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)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,38)(9,37)(10,36)(11,42)(12,41)(13,40)(14,39)(15,45)(16,44)(17,43)(18,49)(19,48)(20,47)(21,46)(22,52)(23,51)(24,50)(25,56)(26,55)(27,54)(28,53)(57,108)(58,107)(59,106)(60,112)(61,111)(62,110)(63,109)(64,101)(65,100)(66,99)(67,105)(68,104)(69,103)(70,102)(71,87)(72,86)(73,85)(74,91)(75,90)(76,89)(77,88)(78,94)(79,93)(80,92)(81,98)(82,97)(83,96)(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)(57,71,64,78)(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), (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(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), (1,104,13,111)(2,105,14,112)(3,99,8,106)(4,100,9,107)(5,101,10,108)(6,102,11,109)(7,103,12,110)(15,92,22,85)(16,93,23,86)(17,94,24,87)(18,95,25,88)(19,96,26,89)(20,97,27,90)(21,98,28,91)(29,71,36,78)(30,72,37,79)(31,73,38,80)(32,74,39,81)(33,75,40,82)(34,76,41,83)(35,77,42,84)(43,64,50,57)(44,65,51,58)(45,66,52,59)(46,67,53,60)(47,68,54,61)(48,69,55,62)(49,70,56,63), (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)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,38)(9,37)(10,36)(11,42)(12,41)(13,40)(14,39)(15,45)(16,44)(17,43)(18,49)(19,48)(20,47)(21,46)(22,52)(23,51)(24,50)(25,56)(26,55)(27,54)(28,53)(57,108)(58,107)(59,106)(60,112)(61,111)(62,110)(63,109)(64,101)(65,100)(66,99)(67,105)(68,104)(69,103)(70,102)(71,87)(72,86)(73,85)(74,91)(75,90)(76,89)(77,88)(78,94)(79,93)(80,92)(81,98)(82,97)(83,96)(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),(57,71,64,78),(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)], [(1,20,13,27),(2,21,14,28),(3,15,8,22),(4,16,9,23),(5,17,10,24),(6,18,11,25),(7,19,12,26),(29,50,36,43),(30,51,37,44),(31,52,38,45),(32,53,39,46),(33,54,40,47),(34,55,41,48),(35,56,42,49),(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)], [(1,104,13,111),(2,105,14,112),(3,99,8,106),(4,100,9,107),(5,101,10,108),(6,102,11,109),(7,103,12,110),(15,92,22,85),(16,93,23,86),(17,94,24,87),(18,95,25,88),(19,96,26,89),(20,97,27,90),(21,98,28,91),(29,71,36,78),(30,72,37,79),(31,73,38,80),(32,74,39,81),(33,75,40,82),(34,76,41,83),(35,77,42,84),(43,64,50,57),(44,65,51,58),(45,66,52,59),(46,67,53,60),(47,68,54,61),(48,69,55,62),(49,70,56,63)], [(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),(2,32),(3,31),(4,30),(5,29),(6,35),(7,34),(8,38),(9,37),(10,36),(11,42),(12,41),(13,40),(14,39),(15,45),(16,44),(17,43),(18,49),(19,48),(20,47),(21,46),(22,52),(23,51),(24,50),(25,56),(26,55),(27,54),(28,53),(57,108),(58,107),(59,106),(60,112),(61,111),(62,110),(63,109),(64,101),(65,100),(66,99),(67,105),(68,104),(69,103),(70,102),(71,87),(72,86),(73,85),(74,91),(75,90),(76,89),(77,88),(78,94),(79,93),(80,92),(81,98),(82,97),(83,96),(84,95)]])`

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14R 28A ··· 28AD order 1 2 2 2 2 2 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 56 2 2 4 4 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 D4.8D4 2- 1+4⋊D7 kernel 2- 1+4⋊D7 C28.10D4 D4⋊2Dic7 C28.23D4 D4⋊D14 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)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 92 98 0 0 0 0 27 0 98 0 0 0 20 73 22 15
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 33 44 98 0 0 0 5 108 11 88 0 0 42 52 52 2 0 0 6 1 90 33
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 98 0 0 0 0 0 21 15 0 0 0 0 39 88 98 0 0 0 38 56 22 15
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 16 73 22 30 0 0 5 27 6 92 0 0 90 41 66 27 0 0 79 80 91 4
,
 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 37 69 0 0 0 0 103 76 0 0 0 0 53 56 22 30 0 0 99 53 63 91

`G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,92,27,20,0,0,0,98,0,73,0,0,0,0,98,22,0,0,0,0,0,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,5,42,6,0,0,44,108,52,1,0,0,98,11,52,90,0,0,0,88,2,33],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,98,21,39,38,0,0,0,15,88,56,0,0,0,0,98,22,0,0,0,0,0,15],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,5,90,79,0,0,73,27,41,80,0,0,22,6,66,91,0,0,30,92,27,4],[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],[17,47,0,0,0,0,66,96,0,0,0,0,0,0,37,103,53,99,0,0,69,76,56,53,0,0,0,0,22,63,0,0,0,0,30,91] >;`

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(448,779);`
`// by ID`

`G=gap.SmallGroup(448,779);`
`# by ID`

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

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

׿
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