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## G = C14.532+ 1+4order 448 = 26·7

### 53rd 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 — C14.532+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C14.532+ 1+4
 Lower central C7 — C2×C14 — C14.532+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C14.532+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, 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=b2d >

Subgroups: 1276 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C422C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×C14, C22.45C24, 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, Q8×C14, C23×D7, C23.11D14, D7×C22⋊C4, C22⋊D28, C22.D28, C4⋊C47D7, D28⋊C4, D14.5D4, C4⋊C4⋊D7, C2×D14⋊C4, C4×C7⋊D4, D143Q8, C28.23D4, C7×C22⋊Q8, C14.532+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, Q82D7, C23×D7, D46D14, C2×Q82D7, D7×C4○D4, C14.532+ 1+4

Smallest permutation representation of C14.532+ 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 57 25 81)(2 58 26 82)(3 59 27 83)(4 60 28 84)(5 61 15 71)(6 62 16 72)(7 63 17 73)(8 64 18 74)(9 65 19 75)(10 66 20 76)(11 67 21 77)(12 68 22 78)(13 69 23 79)(14 70 24 80)(29 90 56 106)(30 91 43 107)(31 92 44 108)(32 93 45 109)(33 94 46 110)(34 95 47 111)(35 96 48 112)(36 97 49 99)(37 98 50 100)(38 85 51 101)(39 86 52 102)(40 87 53 103)(41 88 54 104)(42 89 55 105)
(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 71)(69 72)(70 73)(85 108)(86 109)(87 110)(88 111)(89 112)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)
(1 54 18 34)(2 53 19 33)(3 52 20 32)(4 51 21 31)(5 50 22 30)(6 49 23 29)(7 48 24 42)(8 47 25 41)(9 46 26 40)(10 45 27 39)(11 44 28 38)(12 43 15 37)(13 56 16 36)(14 55 17 35)(57 111 74 88)(58 110 75 87)(59 109 76 86)(60 108 77 85)(61 107 78 98)(62 106 79 97)(63 105 80 96)(64 104 81 95)(65 103 82 94)(66 102 83 93)(67 101 84 92)(68 100 71 91)(69 99 72 90)(70 112 73 89)
(1 34 8 41)(2 35 9 42)(3 36 10 29)(4 37 11 30)(5 38 12 31)(6 39 13 32)(7 40 14 33)(15 51 22 44)(16 52 23 45)(17 53 24 46)(18 54 25 47)(19 55 26 48)(20 56 27 49)(21 43 28 50)(57 88 64 95)(58 89 65 96)(59 90 66 97)(60 91 67 98)(61 92 68 85)(62 93 69 86)(63 94 70 87)(71 108 78 101)(72 109 79 102)(73 110 80 103)(74 111 81 104)(75 112 82 105)(76 99 83 106)(77 100 84 107)```

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

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

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

67 conjugacy classes

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

67 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 C4○D4 D14 D14 D14 D14 2+ 1+4 Q8⋊2D7 D4⋊6D14 D7×C4○D4 kernel C14.532+ 1+4 C23.11D14 D7×C22⋊C4 C22⋊D28 C22.D28 C4⋊C4⋊7D7 D28⋊C4 D14.5D4 C4⋊C4⋊D7 C2×D14⋊C4 C4×C7⋊D4 D14⋊3Q8 C28.23D4 C7×C22⋊Q8 C22⋊Q8 D14 C2×C14 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C14 C22 C2 C2 # reps 1 1 1 1 1 1 1 2 2 1 1 1 1 1 3 4 4 6 9 3 3 1 6 6 6

Matrix representation of C14.532+ 1+4 in GL6(𝔽29)

 28 21 0 0 0 0 17 19 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
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 12 0 0 0 0 0 0 17 0 0 0 0 0 0 17 27 0 0 0 0 28 12
,
 1 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 0 0 0 0 0 17 28
,
 1 8 0 0 0 0 0 28 0 0 0 0 0 0 0 28 0 0 0 0 28 0 0 0 0 0 0 0 12 0 0 0 0 0 1 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 28 0 0 0 0 0 0 0 17 0 0 0 0 0 28 12

`G:=sub<GL(6,GF(29))| [28,17,0,0,0,0,21,19,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,17,28,0,0,0,0,27,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,28],[1,0,0,0,0,0,8,28,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,1,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,17,28,0,0,0,0,0,12] >;`

C14.532+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{53}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.53ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1090);`
`// by ID`

`G=gap.SmallGroup(448,1090);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,297,136,18822]);`
`// Polycyclic`

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

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