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## G = C14.622+ 1+4order 448 = 26·7

### 62nd 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 — C14.622+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C14.622+ 1+4
 Lower central C7 — C2×C14 — C14.622+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.622+ 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, dbd-1=ebe=a7b, dcd-1=ece=a7c, ede=b2d >

Subgroups: 1228 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, 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, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, Dic14, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.45C24, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23.D14, D7×C22⋊C4, D14.D4, C4⋊C47D7, D14⋊Q8, C4×C7⋊D4, C28.17D4, C23⋊D14, C7×C22.D4, C14.622+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, C23×D7, D46D14, D7×C4○D4, C14.622+ 1+4

Smallest permutation representation of C14.622+ 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 66 28 84)(2 67 15 71)(3 68 16 72)(4 69 17 73)(5 70 18 74)(6 57 19 75)(7 58 20 76)(8 59 21 77)(9 60 22 78)(10 61 23 79)(11 62 24 80)(12 63 25 81)(13 64 26 82)(14 65 27 83)(29 90 56 108)(30 91 43 109)(31 92 44 110)(32 93 45 111)(33 94 46 112)(34 95 47 99)(35 96 48 100)(36 97 49 101)(37 98 50 102)(38 85 51 103)(39 86 52 104)(40 87 53 105)(41 88 54 106)(42 89 55 107)
(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 82)(58 83)(59 84)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 99)(96 100)(97 101)(98 102)
(1 56 28 29)(2 55 15 42)(3 54 16 41)(4 53 17 40)(5 52 18 39)(6 51 19 38)(7 50 20 37)(8 49 21 36)(9 48 22 35)(10 47 23 34)(11 46 24 33)(12 45 25 32)(13 44 26 31)(14 43 27 30)(57 110 75 92)(58 109 76 91)(59 108 77 90)(60 107 78 89)(61 106 79 88)(62 105 80 87)(63 104 81 86)(64 103 82 85)(65 102 83 98)(66 101 84 97)(67 100 71 96)(68 99 72 95)(69 112 73 94)(70 111 74 93)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 85)(69 86)(70 87)(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), (1,66,28,84)(2,67,15,71)(3,68,16,72)(4,69,17,73)(5,70,18,74)(6,57,19,75)(7,58,20,76)(8,59,21,77)(9,60,22,78)(10,61,23,79)(11,62,24,80)(12,63,25,81)(13,64,26,82)(14,65,27,83)(29,90,56,108)(30,91,43,109)(31,92,44,110)(32,93,45,111)(33,94,46,112)(34,95,47,99)(35,96,48,100)(36,97,49,101)(37,98,50,102)(38,85,51,103)(39,86,52,104)(40,87,53,105)(41,88,54,106)(42,89,55,107), (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)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,99)(96,100)(97,101)(98,102), (1,56,28,29)(2,55,15,42)(3,54,16,41)(4,53,17,40)(5,52,18,39)(6,51,19,38)(7,50,20,37)(8,49,21,36)(9,48,22,35)(10,47,23,34)(11,46,24,33)(12,45,25,32)(13,44,26,31)(14,43,27,30)(57,110,75,92)(58,109,76,91)(59,108,77,90)(60,107,78,89)(61,106,79,88)(62,105,80,87)(63,104,81,86)(64,103,82,85)(65,102,83,98)(66,101,84,97)(67,100,71,96)(68,99,72,95)(69,112,73,94)(70,111,74,93), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,85)(69,86)(70,87)(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), (1,66,28,84)(2,67,15,71)(3,68,16,72)(4,69,17,73)(5,70,18,74)(6,57,19,75)(7,58,20,76)(8,59,21,77)(9,60,22,78)(10,61,23,79)(11,62,24,80)(12,63,25,81)(13,64,26,82)(14,65,27,83)(29,90,56,108)(30,91,43,109)(31,92,44,110)(32,93,45,111)(33,94,46,112)(34,95,47,99)(35,96,48,100)(36,97,49,101)(37,98,50,102)(38,85,51,103)(39,86,52,104)(40,87,53,105)(41,88,54,106)(42,89,55,107), (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)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,99)(96,100)(97,101)(98,102), (1,56,28,29)(2,55,15,42)(3,54,16,41)(4,53,17,40)(5,52,18,39)(6,51,19,38)(7,50,20,37)(8,49,21,36)(9,48,22,35)(10,47,23,34)(11,46,24,33)(12,45,25,32)(13,44,26,31)(14,43,27,30)(57,110,75,92)(58,109,76,91)(59,108,77,90)(60,107,78,89)(61,106,79,88)(62,105,80,87)(63,104,81,86)(64,103,82,85)(65,102,83,98)(66,101,84,97)(67,100,71,96)(68,99,72,95)(69,112,73,94)(70,111,74,93), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,85)(69,86)(70,87)(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)], [(1,66,28,84),(2,67,15,71),(3,68,16,72),(4,69,17,73),(5,70,18,74),(6,57,19,75),(7,58,20,76),(8,59,21,77),(9,60,22,78),(10,61,23,79),(11,62,24,80),(12,63,25,81),(13,64,26,82),(14,65,27,83),(29,90,56,108),(30,91,43,109),(31,92,44,110),(32,93,45,111),(33,94,46,112),(34,95,47,99),(35,96,48,100),(36,97,49,101),(37,98,50,102),(38,85,51,103),(39,86,52,104),(40,87,53,105),(41,88,54,106),(42,89,55,107)], [(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),(57,82),(58,83),(59,84),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,99),(96,100),(97,101),(98,102)], [(1,56,28,29),(2,55,15,42),(3,54,16,41),(4,53,17,40),(5,52,18,39),(6,51,19,38),(7,50,20,37),(8,49,21,36),(9,48,22,35),(10,47,23,34),(11,46,24,33),(12,45,25,32),(13,44,26,31),(14,43,27,30),(57,110,75,92),(58,109,76,91),(59,108,77,90),(60,107,78,89),(61,106,79,88),(62,105,80,87),(63,104,81,86),(64,103,82,85),(65,102,83,98),(66,101,84,97),(67,100,71,96),(68,99,72,95),(69,112,73,94),(70,111,74,93)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,85),(69,86),(70,87),(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)]])`

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 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 4 4 4 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 14 14 14 14 2 2 2 2 4 4 4 14 14 14 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

67 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D14 2+ 1+4 D4⋊6D14 D7×C4○D4 kernel C14.622+ 1+4 C23.D14 D7×C22⋊C4 D14.D4 C4⋊C4⋊7D7 D14⋊Q8 C4×C7⋊D4 C28.17D4 C23⋊D14 C7×C22.D4 C22.D4 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C2 C2 # reps 1 2 2 2 2 2 2 1 1 1 3 8 9 6 3 3 1 6 12

Matrix representation of C14.622+ 1+4 in GL6(𝔽29)

 8 25 0 0 0 0 24 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 17 0 0 0 0 0 0 0 12 0 0 0 0 0 18 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 16 28
,
 0 18 0 0 0 0 21 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 28 0 0 0 0 7 21
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 28 0 0 0 0 5 21

`G:=sub<GL(6,GF(29))| [8,24,0,0,0,0,25,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,17,0,0,0,0,12,0,0,0,0,0,0,0,12,18,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,16,0,0,0,0,0,28],[0,21,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,7,0,0,0,0,28,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,5,0,0,0,0,28,21] >;`

C14.622+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{62}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.62ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1112);`
`// by ID`

`G=gap.SmallGroup(448,1112);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,570,18822]);`
`// 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,d*b*d^-1=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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