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## G = C14.372+ 1+4order 448 = 26·7

### 37th 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 — C14.372+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D4×D7 — C14.372+ 1+4
 Lower central C7 — C2×C14 — C14.372+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C14.372+ 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=b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 2124 in 346 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C233D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, C23×D7, C22⋊D28, D14.D4, D14.5D4, C2×D14⋊C4, C287D4, C23⋊D14, Dic7⋊D4, C7×C4⋊D4, C2×D4×D7, C14.372+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C22×D7, C233D4, D4×D7, C23×D7, C2×D4×D7, D46D14, D48D14, C14.372+ 1+4

Smallest permutation representation of C14.372+ 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 19 81)(2 66 20 82)(3 67 21 83)(4 68 22 84)(5 69 23 71)(6 70 24 72)(7 57 25 73)(8 58 26 74)(9 59 27 75)(10 60 28 76)(11 61 15 77)(12 62 16 78)(13 63 17 79)(14 64 18 80)(29 96 49 107)(30 97 50 108)(31 98 51 109)(32 85 52 110)(33 86 53 111)(34 87 54 112)(35 88 55 99)(36 89 56 100)(37 90 43 101)(38 91 44 102)(39 92 45 103)(40 93 46 104)(41 94 47 105)(42 95 48 106)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(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 80)(58 81)(59 82)(60 83)(61 84)(62 71)(63 72)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(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 26 29)(2 55 27 42)(3 54 28 41)(4 53 15 40)(5 52 16 39)(6 51 17 38)(7 50 18 37)(8 49 19 36)(9 48 20 35)(10 47 21 34)(11 46 22 33)(12 45 23 32)(13 44 24 31)(14 43 25 30)(57 101 80 97)(58 100 81 96)(59 99 82 95)(60 112 83 94)(61 111 84 93)(62 110 71 92)(63 109 72 91)(64 108 73 90)(65 107 74 89)(66 106 75 88)(67 105 76 87)(68 104 77 86)(69 103 78 85)(70 102 79 98)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 52 22 45)(16 53 23 46)(17 54 24 47)(18 55 25 48)(19 56 26 49)(20 43 27 50)(21 44 28 51)(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)(71 111 78 104)(72 112 79 105)(73 99 80 106)(74 100 81 107)(75 101 82 108)(76 102 83 109)(77 103 84 110)```

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 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 C2 D4 D7 D14 D14 D14 D14 2+ 1+4 D4×D7 D4⋊6D14 D4⋊8D14 kernel C14.372+ 1+4 C22⋊D28 D14.D4 D14.5D4 C2×D14⋊C4 C28⋊7D4 C23⋊D14 Dic7⋊D4 C7×C4⋊D4 C2×D4×D7 C22×D7 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C22 C2 C2 # reps 1 2 2 2 1 1 2 2 1 2 4 3 6 3 3 9 2 6 6 6

Matrix representation of C14.372+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 10 10 0 0 0 0 19 22 0 0 0 0 0 0 9 10 0 0 0 0 9 23
,
 22 3 0 0 0 0 13 7 0 0 0 0 0 0 28 0 5 14 0 0 0 28 15 0 0 0 0 25 1 0 0 0 4 18 0 1
,
 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 4 28 0 0 0 25 11 0 28
,
 13 24 0 0 0 0 5 16 0 0 0 0 0 0 20 15 0 0 0 0 10 9 0 0 0 0 0 0 22 19 0 0 0 0 5 7
,
 16 5 0 0 0 0 24 13 0 0 0 0 0 0 20 15 0 0 0 0 14 9 0 0 0 0 0 0 4 15 0 0 0 0 28 25

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,9,9,0,0,0,0,10,23],[22,13,0,0,0,0,3,7,0,0,0,0,0,0,28,0,0,4,0,0,0,28,25,18,0,0,5,15,1,0,0,0,14,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,25,0,0,0,1,4,11,0,0,0,0,28,0,0,0,0,0,0,28],[13,5,0,0,0,0,24,16,0,0,0,0,0,0,20,10,0,0,0,0,15,9,0,0,0,0,0,0,22,5,0,0,0,0,19,7],[16,24,0,0,0,0,5,13,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,4,28,0,0,0,0,15,25] >;`

C14.372+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{37}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.37ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1058);`
`// by ID`

`G=gap.SmallGroup(448,1058);`
`# by ID`

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