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## G = C14.612+ 1+4order 448 = 26·7

### 61st 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 — C14.612+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C14.612+ 1+4
 Lower central C7 — C2×C14 — C14.612+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.612+ 1+4
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe=a7b, dcd-1=ece=a7c, ede=a7b2d >

Subgroups: 1388 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, 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×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23.D14, D7×C22⋊C4, C22⋊D28, D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, D142Q8, C4⋊C4⋊D7, C4×C7⋊D4, C287D4, C23⋊D14, C282D4, C7×C22.D4, C14.612+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C23×D7, D46D14, D7×C4○D4, D48D14, C14.612+ 1+4

Smallest permutation representation of C14.612+ 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 67 20 71)(2 68 21 72)(3 69 22 73)(4 70 23 74)(5 57 24 75)(6 58 25 76)(7 59 26 77)(8 60 27 78)(9 61 28 79)(10 62 15 80)(11 63 16 81)(12 64 17 82)(13 65 18 83)(14 66 19 84)(29 88 50 100)(30 89 51 101)(31 90 52 102)(32 91 53 103)(33 92 54 104)(34 93 55 105)(35 94 56 106)(36 95 43 107)(37 96 44 108)(38 97 45 109)(39 98 46 110)(40 85 47 111)(41 86 48 112)(42 87 49 99)
(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)(85 111)(86 112)(87 99)(88 100)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)
(1 45 27 31)(2 44 28 30)(3 43 15 29)(4 56 16 42)(5 55 17 41)(6 54 18 40)(7 53 19 39)(8 52 20 38)(9 51 21 37)(10 50 22 36)(11 49 23 35)(12 48 24 34)(13 47 25 33)(14 46 26 32)(57 112 82 93)(58 111 83 92)(59 110 84 91)(60 109 71 90)(61 108 72 89)(62 107 73 88)(63 106 74 87)(64 105 75 86)(65 104 76 85)(66 103 77 98)(67 102 78 97)(68 101 79 96)(69 100 80 95)(70 99 81 94)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(57 94)(58 95)(59 96)(60 97)(61 98)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(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), (1,67,20,71)(2,68,21,72)(3,69,22,73)(4,70,23,74)(5,57,24,75)(6,58,25,76)(7,59,26,77)(8,60,27,78)(9,61,28,79)(10,62,15,80)(11,63,16,81)(12,64,17,82)(13,65,18,83)(14,66,19,84)(29,88,50,100)(30,89,51,101)(31,90,52,102)(32,91,53,103)(33,92,54,104)(34,93,55,105)(35,94,56,106)(36,95,43,107)(37,96,44,108)(38,97,45,109)(39,98,46,110)(40,85,47,111)(41,86,48,112)(42,87,49,99), (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)(85,111)(86,112)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110), (1,45,27,31)(2,44,28,30)(3,43,15,29)(4,56,16,42)(5,55,17,41)(6,54,18,40)(7,53,19,39)(8,52,20,38)(9,51,21,37)(10,50,22,36)(11,49,23,35)(12,48,24,34)(13,47,25,33)(14,46,26,32)(57,112,82,93)(58,111,83,92)(59,110,84,91)(60,109,71,90)(61,108,72,89)(62,107,73,88)(63,106,74,87)(64,105,75,86)(65,104,76,85)(66,103,77,98)(67,102,78,97)(68,101,79,96)(69,100,80,95)(70,99,81,94), (1,38)(2,39)(3,40)(4,41)(5,42)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(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), (1,67,20,71)(2,68,21,72)(3,69,22,73)(4,70,23,74)(5,57,24,75)(6,58,25,76)(7,59,26,77)(8,60,27,78)(9,61,28,79)(10,62,15,80)(11,63,16,81)(12,64,17,82)(13,65,18,83)(14,66,19,84)(29,88,50,100)(30,89,51,101)(31,90,52,102)(32,91,53,103)(33,92,54,104)(34,93,55,105)(35,94,56,106)(36,95,43,107)(37,96,44,108)(38,97,45,109)(39,98,46,110)(40,85,47,111)(41,86,48,112)(42,87,49,99), (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)(85,111)(86,112)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110), (1,45,27,31)(2,44,28,30)(3,43,15,29)(4,56,16,42)(5,55,17,41)(6,54,18,40)(7,53,19,39)(8,52,20,38)(9,51,21,37)(10,50,22,36)(11,49,23,35)(12,48,24,34)(13,47,25,33)(14,46,26,32)(57,112,82,93)(58,111,83,92)(59,110,84,91)(60,109,71,90)(61,108,72,89)(62,107,73,88)(63,106,74,87)(64,105,75,86)(65,104,76,85)(66,103,77,98)(67,102,78,97)(68,101,79,96)(69,100,80,95)(70,99,81,94), (1,38)(2,39)(3,40)(4,41)(5,42)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(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)], [(1,67,20,71),(2,68,21,72),(3,69,22,73),(4,70,23,74),(5,57,24,75),(6,58,25,76),(7,59,26,77),(8,60,27,78),(9,61,28,79),(10,62,15,80),(11,63,16,81),(12,64,17,82),(13,65,18,83),(14,66,19,84),(29,88,50,100),(30,89,51,101),(31,90,52,102),(32,91,53,103),(33,92,54,104),(34,93,55,105),(35,94,56,106),(36,95,43,107),(37,96,44,108),(38,97,45,109),(39,98,46,110),(40,85,47,111),(41,86,48,112),(42,87,49,99)], [(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),(85,111),(86,112),(87,99),(88,100),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110)], [(1,45,27,31),(2,44,28,30),(3,43,15,29),(4,56,16,42),(5,55,17,41),(6,54,18,40),(7,53,19,39),(8,52,20,38),(9,51,21,37),(10,50,22,36),(11,49,23,35),(12,48,24,34),(13,47,25,33),(14,46,26,32),(57,112,82,93),(58,111,83,92),(59,110,84,91),(60,109,71,90),(61,108,72,89),(62,107,73,88),(63,106,74,87),(64,105,75,86),(65,104,76,85),(66,103,77,98),(67,102,78,97),(68,101,79,96),(69,100,80,95),(70,99,81,94)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,54),(16,55),(17,56),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(57,94),(58,95),(59,96),(60,97),(61,98),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(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)]])`

64 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 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 7 7 7 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 14 14 28 28 2 2 4 4 4 4 14 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 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 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D14 2+ 1+4 D4⋊6D14 D7×C4○D4 D4⋊8D14 kernel C14.612+ 1+4 C23.D14 D7×C22⋊C4 C22⋊D28 D14⋊D4 Dic7.D4 D28⋊C4 D14.5D4 D14⋊2Q8 C4⋊C4⋊D7 C4×C7⋊D4 C28⋊7D4 C23⋊D14 C28⋊2D4 C7×C22.D4 C22.D4 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 3 4 9 6 3 3 2 6 6 6

Matrix representation of C14.612+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 19 19 0 0 0 0 10 7 0 0 0 0 0 0 19 19 0 0 0 0 10 7
,
 17 0 0 0 0 0 23 12 0 0 0 0 0 0 8 6 13 17 0 0 23 21 12 16 0 0 8 6 21 23 0 0 23 21 6 8
,
 1 0 0 0 0 0 15 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 28 0 0 0 0 1 0 28
,
 23 24 0 0 0 0 7 6 0 0 0 0 0 0 28 0 2 0 0 0 22 1 14 27 0 0 28 0 1 0 0 0 22 1 7 28
,
 23 24 0 0 0 0 7 6 0 0 0 0 0 0 1 0 27 0 0 0 0 1 0 27 0 0 0 0 28 0 0 0 0 0 0 28

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,19,10,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,19,7],[17,23,0,0,0,0,0,12,0,0,0,0,0,0,8,23,8,23,0,0,6,21,6,21,0,0,13,12,21,6,0,0,17,16,23,8],[1,15,0,0,0,0,0,28,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,28,0,0,0,0,0,0,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,28,22,28,22,0,0,0,1,0,1,0,0,2,14,1,7,0,0,0,27,0,28],[23,7,0,0,0,0,24,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,27,0,28,0,0,0,0,27,0,28] >;`

C14.612+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{61}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.61ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1110);`
`// by ID`

`G=gap.SmallGroup(448,1110);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,18822]);`
`// Polycyclic`

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

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