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## G = C14.C25order 448 = 26·7

### 14th non-split extension by C14 of C25 acting via C25/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C14.C25
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — D7×C4○D4 — C14.C25
 Lower central C7 — C14 — C14.C25
 Upper central C1 — C4 — C2×C4○D4

Generators and relations for C14.C25
G = < a,b,c,d,e,f | a14=b2=c2=e2=f2=1, d2=a7, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a7b, cd=dc, ece=a7c, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 3156 in 810 conjugacy classes, 443 normal (17 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, C14, C14, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2- 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C2.C25, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, C2×C7⋊D4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C2×C4○D28, D46D14, Q8.10D14, D7×C4○D4, D48D14, D4.10D14, C14×C4○D4, C14.C25
Quotients: C1, C2, C22, C23, D7, C24, D14, C25, C22×D7, C2.C25, C23×D7, D7×C24, C14.C25

Smallest permutation representation of C14.C25
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)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 16)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(55 56)(57 67)(58 66)(59 65)(60 64)(61 63)(68 70)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 84)(85 87)(88 98)(89 97)(90 96)(91 95)(92 94)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(111 112)
(1 69)(2 70)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 71)(25 72)(26 73)(27 74)(28 75)(29 97)(30 98)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 39 8 32)(2 40 9 33)(3 41 10 34)(4 42 11 35)(5 29 12 36)(6 30 13 37)(7 31 14 38)(15 55 22 48)(16 56 23 49)(17 43 24 50)(18 44 25 51)(19 45 26 52)(20 46 27 53)(21 47 28 54)(57 95 64 88)(58 96 65 89)(59 97 66 90)(60 98 67 91)(61 85 68 92)(62 86 69 93)(63 87 70 94)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)
(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 15)(12 16)(13 17)(14 18)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(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)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)(91 99)(92 100)(93 101)(94 102)(95 103)(96 104)(97 105)(98 106)```

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

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

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

94 conjugacy classes

 class 1 2A 2B ··· 2H 2I ··· 2P 4A 4B 4C ··· 4I 4J ··· 4Q 7A 7B 7C 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 ··· 2 2 ··· 2 4 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 ··· 2 14 ··· 14 1 1 2 ··· 2 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

94 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D7 D14 D14 D14 D14 C2.C25 C14.C25 kernel C14.C25 C2×C4○D28 D4⋊6D14 Q8.10D14 D7×C4○D4 D4⋊8D14 D4.10D14 C14×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C7 C1 # reps 1 6 6 2 8 4 4 1 3 9 9 3 24 2 12

Matrix representation of C14.C25 in GL4(𝔽29) generated by

 4 4 0 0 25 18 0 0 0 0 4 4 0 0 25 18
,
 1 0 0 0 18 28 0 0 0 0 1 0 0 0 18 28
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 18 27 0 0 2 11 0 0 0 0 18 27 0 0 2 11
`G:=sub<GL(4,GF(29))| [4,25,0,0,4,18,0,0,0,0,4,25,0,0,4,18],[1,18,0,0,0,28,0,0,0,0,1,18,0,0,0,28],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[18,2,0,0,27,11,0,0,0,0,18,2,0,0,27,11] >;`

C14.C25 in GAP, Magma, Sage, TeX

`C_{14}.C_2^5`
`% in TeX`

`G:=Group("C14.C2^5");`
`// GroupNames label`

`G:=SmallGroup(448,1378);`
`// by ID`

`G=gap.SmallGroup(448,1378);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,1684,18822]);`
`// Polycyclic`

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

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