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## G = C14.512+ 1+4order 448 = 26·7

### 51st 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 — C14.512+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C14.512+ 1+4
 Lower central C7 — C2×C14 — C14.512+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C14.512+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >

Subgroups: 1244 in 242 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22⋊Q8, C22⋊Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×C14, C232Q8, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C22×Dic7, C22×C28, Q8×C14, C23×D7, C22⋊Dic14, D7×C22⋊C4, D14⋊Q8, D142Q8, C28.48D4, C2×D14⋊C4, D143Q8, C7×C22⋊Q8, C14.512+ 1+4
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, 2+ 1+4, C22×D7, C232Q8, Q8×D7, C23×D7, D46D14, C2×Q8×D7, D48D14, C14.512+ 1+4

Smallest permutation representation of C14.512+ 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 65 25 74)(2 66 26 75)(3 67 27 76)(4 68 28 77)(5 69 15 78)(6 70 16 79)(7 57 17 80)(8 58 18 81)(9 59 19 82)(10 60 20 83)(11 61 21 84)(12 62 22 71)(13 63 23 72)(14 64 24 73)(29 98 45 102)(30 85 46 103)(31 86 47 104)(32 87 48 105)(33 88 49 106)(34 89 50 107)(35 90 51 108)(36 91 52 109)(37 92 53 110)(38 93 54 111)(39 94 55 112)(40 95 56 99)(41 96 43 100)(42 97 44 101)
(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 71)(70 72)(85 110)(86 111)(87 112)(88 99)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)
(1 44 25 42)(2 43 26 41)(3 56 27 40)(4 55 28 39)(5 54 15 38)(6 53 16 37)(7 52 17 36)(8 51 18 35)(9 50 19 34)(10 49 20 33)(11 48 21 32)(12 47 22 31)(13 46 23 30)(14 45 24 29)(57 102 80 98)(58 101 81 97)(59 100 82 96)(60 99 83 95)(61 112 84 94)(62 111 71 93)(63 110 72 92)(64 109 73 91)(65 108 74 90)(66 107 75 89)(67 106 76 88)(68 105 77 87)(69 104 78 86)(70 103 79 85)
(1 42 8 35)(2 29 9 36)(3 30 10 37)(4 31 11 38)(5 32 12 39)(6 33 13 40)(7 34 14 41)(15 48 22 55)(16 49 23 56)(17 50 24 43)(18 51 25 44)(19 52 26 45)(20 53 27 46)(21 54 28 47)(57 96 64 89)(58 97 65 90)(59 98 66 91)(60 85 67 92)(61 86 68 93)(62 87 69 94)(63 88 70 95)(71 105 78 112)(72 106 79 99)(73 107 80 100)(74 108 81 101)(75 109 82 102)(76 110 83 103)(77 111 84 104)```

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D7 D14 D14 D14 D14 2+ 1+4 Q8×D7 D4⋊6D14 D4⋊8D14 kernel C14.512+ 1+4 C22⋊Dic14 D7×C22⋊C4 D14⋊Q8 D14⋊2Q8 C28.48D4 C2×D14⋊C4 D14⋊3Q8 C7×C22⋊Q8 C22×D7 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C14 C22 C2 C2 # reps 1 2 2 4 2 1 1 2 1 4 3 6 9 3 3 2 6 6 6

Matrix representation of C14.512+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 18 18 0 26 0 0 8 0 11 18 0 0 0 0 26 21 0 0 0 0 8 21
,
 2 13 0 0 0 0 13 27 0 0 0 0 0 0 0 25 21 20 0 0 0 19 15 15 0 0 28 27 5 5 0 0 1 3 5 5
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 8 8 0 0 0 1 20 20 0 0 0 0 28 0 0 0 0 0 0 28
,
 0 1 0 0 0 0 28 0 0 0 0 0 0 0 22 19 20 0 0 0 5 7 25 4 0 0 0 0 5 2 0 0 0 0 16 24
,
 0 28 0 0 0 0 1 0 0 0 0 0 0 0 11 22 0 6 0 0 13 18 7 22 0 0 0 0 24 16 0 0 0 0 13 5

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,18,8,0,0,0,0,18,0,0,0,0,0,0,11,26,8,0,0,26,18,21,21],[2,13,0,0,0,0,13,27,0,0,0,0,0,0,0,0,28,1,0,0,25,19,27,3,0,0,21,15,5,5,0,0,20,15,5,5],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,8,20,28,0,0,0,8,20,0,28],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,22,5,0,0,0,0,19,7,0,0,0,0,20,25,5,16,0,0,0,4,2,24],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,11,13,0,0,0,0,22,18,0,0,0,0,0,7,24,13,0,0,6,22,16,5] >;`

C14.512+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{51}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.51ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1087);`
`// by ID`

`G=gap.SmallGroup(448,1087);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,297,136,18822]);`
`// Polycyclic`

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

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