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## G = C14.682+ 1+4order 448 = 26·7

### 68th 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 — C14.682+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22⋊D28 — C14.682+ 1+4
 Lower central C7 — C2×C14 — C14.682+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.682+ 1+4
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, bd=db, ebe=a7b, dcd-1=ece=a7c, ede=b2d >

Subgroups: 1548 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, 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, C22.D4, C22.D4, C422C2, C41D4, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, 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×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C22⋊D28, D14.D4, D14⋊D4, C4⋊D28, C4⋊C4⋊D7, C287D4, C23⋊D14, C28⋊D4, C7×C22.D4, C14.682+ 1+4
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C22.54C24, C23×D7, D46D14, D48D14, C14.682+ 1+4

Smallest permutation representation of C14.682+ 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 69 20 76)(2 70 21 77)(3 57 22 78)(4 58 23 79)(5 59 24 80)(6 60 25 81)(7 61 26 82)(8 62 27 83)(9 63 28 84)(10 64 15 71)(11 65 16 72)(12 66 17 73)(13 67 18 74)(14 68 19 75)(29 86 47 110)(30 87 48 111)(31 88 49 112)(32 89 50 99)(33 90 51 100)(34 91 52 101)(35 92 53 102)(36 93 54 103)(37 94 55 104)(38 95 56 105)(39 96 43 106)(40 97 44 107)(41 98 45 108)(42 85 46 109)
(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 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 109)(86 110)(87 111)(88 112)(89 99)(90 100)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)
(1 33 20 51)(2 32 21 50)(3 31 22 49)(4 30 23 48)(5 29 24 47)(6 42 25 46)(7 41 26 45)(8 40 27 44)(9 39 28 43)(10 38 15 56)(11 37 16 55)(12 36 17 54)(13 35 18 53)(14 34 19 52)(57 88 78 112)(58 87 79 111)(59 86 80 110)(60 85 81 109)(61 98 82 108)(62 97 83 107)(63 96 84 106)(64 95 71 105)(65 94 72 104)(66 93 73 103)(67 92 74 102)(68 91 75 101)(69 90 76 100)(70 89 77 99)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 29)(12 30)(13 31)(14 32)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 43)(27 44)(28 45)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)(81 112)(82 99)(83 100)(84 101)```

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 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 D7 D14 D14 D14 D14 2+ 1+4 D4⋊6D14 D4⋊8D14 kernel C14.682+ 1+4 C22⋊D28 D14.D4 D14⋊D4 C4⋊D28 C4⋊C4⋊D7 C28⋊7D4 C23⋊D14 C28⋊D4 C7×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C2 C2 # reps 1 2 2 2 2 2 2 1 1 1 3 9 6 3 3 3 6 12

Matrix representation of C14.682+ 1+4 in GL8(𝔽29)

 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 7 22 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 7 22
,
 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 8 5 0 0 0 0 0 0 16 21 0 0 0 0 21 24 0 0 0 0 0 0 13 8 0 0
,
 1 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 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
,
 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 0 0 0 17 19 0 0 0 0 10 1 0 0 0 0 0 0 17 19 0 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 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))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,4,22,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,4,22],[0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,21,13,0,0,0,0,0,0,24,8,0,0,0,0,8,16,0,0,0,0,0,0,5,21,0,0],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,10,17,0,0,0,0,0,0,1,19,0,0,0,0,10,17,0,0,0,0,0,0,1,19,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C14.682+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{68}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.68ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1119);`
`// by ID`

`G=gap.SmallGroup(448,1119);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,219,184,1571,570,18822]);`
`// Polycyclic`

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

׿
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