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G = C14.482+ 1+4order 448 = 26·7

48th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.482+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C23⋊D14 — C14.482+ 1+4
 Lower central C7 — C2×C14 — C14.482+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C14.482+ 1+4
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >

Subgroups: 1452 in 252 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C422C2, C41D4, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.54C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23.D14, C22⋊D28, D14⋊D4, C22.D28, D14.5D4, C4⋊C4⋊D7, C23.23D14, C287D4, C23⋊D14, C282D4, Dic7⋊D4, C28⋊D4, C7×C4⋊D4, C14.482+ 1+4
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C22.54C24, C23×D7, D46D14, D48D14, C14.482+ 1+4

Smallest permutation representation of C14.482+ 1+4
On 112 points
Generators in S112
```(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 70 18 82)(2 57 19 83)(3 58 20 84)(4 59 21 71)(5 60 22 72)(6 61 23 73)(7 62 24 74)(8 63 25 75)(9 64 26 76)(10 65 27 77)(11 66 28 78)(12 67 15 79)(13 68 16 80)(14 69 17 81)(29 85 49 105)(30 86 50 106)(31 87 51 107)(32 88 52 108)(33 89 53 109)(34 90 54 110)(35 91 55 111)(36 92 56 112)(37 93 43 99)(38 94 44 100)(39 95 45 101)(40 96 46 102)(41 97 47 103)(42 98 48 104)
(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 83)(58 84)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(85 112)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)
(1 35 25 48)(2 34 26 47)(3 33 27 46)(4 32 28 45)(5 31 15 44)(6 30 16 43)(7 29 17 56)(8 42 18 55)(9 41 19 54)(10 40 20 53)(11 39 21 52)(12 38 22 51)(13 37 23 50)(14 36 24 49)(57 97 76 110)(58 96 77 109)(59 95 78 108)(60 94 79 107)(61 93 80 106)(62 92 81 105)(63 91 82 104)(64 90 83 103)(65 89 84 102)(66 88 71 101)(67 87 72 100)(68 86 73 99)(69 85 74 112)(70 98 75 111)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 52)(16 53)(17 54)(18 55)(19 56)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 99)```

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4I 7A 7B 7C 14A ··· 14I 14J ··· 14O 14P ··· 14U 28A ··· 28L 28M ··· 28R order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 4 28 28 28 4 4 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4 8 ··· 8

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 D14 D14 D14 D14 2+ 1+4 D4⋊6D14 D4⋊8D14 kernel C14.482+ 1+4 C23.D14 C22⋊D28 D14⋊D4 C22.D28 D14.5D4 C4⋊C4⋊D7 C23.23D14 C28⋊7D4 C23⋊D14 C28⋊2D4 Dic7⋊D4 C28⋊D4 C7×C4⋊D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 2 1 1 3 6 3 3 9 3 12 6

Matrix representation of C14.482+ 1+4 in GL8(𝔽29)

 27 13 21 6 0 0 0 0 8 20 0 8 0 0 0 0 0 0 26 21 0 0 0 0 0 0 8 21 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 21 3 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 21 3
,
 11 14 23 26 0 0 0 0 18 16 11 22 0 0 0 0 0 1 14 18 0 0 0 0 28 8 11 17 0 0 0 0 0 0 0 0 0 0 27 11 0 0 0 0 0 0 18 2 0 0 0 0 27 11 0 0 0 0 0 0 18 2 0 0
,
 1 0 28 22 0 0 0 0 0 1 28 7 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 8 18 2 4 0 0 0 0 27 21 24 12 0 0 0 0 0 0 5 2 0 0 0 0 0 0 16 24 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 28 0 0 0 0 1 0 0 0 0 0 0 0 3 28 0 0
,
 7 26 13 12 0 0 0 0 16 22 0 16 0 0 0 0 0 0 5 13 0 0 0 0 0 0 16 24 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(29))| [27,8,0,0,0,0,0,0,13,20,0,0,0,0,0,0,21,0,26,8,0,0,0,0,6,8,21,21,0,0,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3,0,0,0,0,0,0,0,0,8,21,0,0,0,0,0,0,8,3],[11,18,0,28,0,0,0,0,14,16,1,8,0,0,0,0,23,11,14,11,0,0,0,0,26,22,18,17,0,0,0,0,0,0,0,0,0,0,27,18,0,0,0,0,0,0,11,2,0,0,0,0,27,18,0,0,0,0,0,0,11,2,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,28,28,0,0,0,0,0,22,7,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[8,27,0,0,0,0,0,0,18,21,0,0,0,0,0,0,2,24,5,16,0,0,0,0,4,12,2,24,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,28,0,0,0,0,1,3,0,0,0,0,0,0,0,28,0,0],[7,16,0,0,0,0,0,0,26,22,0,0,0,0,0,0,13,0,5,16,0,0,0,0,12,16,13,24,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C14.482+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{48}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.48ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1073);`
`// by ID`

`G=gap.SmallGroup(448,1073);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,1571,570,297,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;`
`// generators/relations`

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