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## G = C14.1462+ 1+4order 448 = 26·7

### 55th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.1462+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22×D28 — C14.1462+ 1+4
 Lower central C7 — C2×C14 — C14.1462+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C14.1462+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, dbd-1=a7b, be=eb, cd=dc, ece=a7c, ede=a7b2d >

Subgroups: 1876 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C41D4, C22×D4, C2×C4○D4, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22×C14, C22.29C24, C4×Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C2×D28, C2×D28, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23×D7, C23.21D14, C287D4, C23⋊D14, C28⋊D4, C28.23D4, C22×D28, C14×C4○D4, C14.1462+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C7⋊D4, C22×D7, C22.29C24, C2×C7⋊D4, C23×D7, D48D14, C22×C7⋊D4, C14.1462+ 1+4

Smallest permutation representation of C14.1462+ 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 95 81 59)(2 96 82 60)(3 97 83 61)(4 98 84 62)(5 85 71 63)(6 86 72 64)(7 87 73 65)(8 88 74 66)(9 89 75 67)(10 90 76 68)(11 91 77 69)(12 92 78 70)(13 93 79 57)(14 94 80 58)(15 44 100 38)(16 45 101 39)(17 46 102 40)(18 47 103 41)(19 48 104 42)(20 49 105 29)(21 50 106 30)(22 51 107 31)(23 52 108 32)(24 53 109 33)(25 54 110 34)(26 55 111 35)(27 56 112 36)(28 43 99 37)
(1 59 8 66)(2 58 9 65)(3 57 10 64)(4 70 11 63)(5 69 12 62)(6 68 13 61)(7 67 14 60)(15 33 22 40)(16 32 23 39)(17 31 24 38)(18 30 25 37)(19 29 26 36)(20 42 27 35)(21 41 28 34)(43 103 50 110)(44 102 51 109)(45 101 52 108)(46 100 53 107)(47 99 54 106)(48 112 55 105)(49 111 56 104)(71 91 78 98)(72 90 79 97)(73 89 80 96)(74 88 81 95)(75 87 82 94)(76 86 83 93)(77 85 84 92)
(1 101 74 23)(2 100 75 22)(3 99 76 21)(4 112 77 20)(5 111 78 19)(6 110 79 18)(7 109 80 17)(8 108 81 16)(9 107 82 15)(10 106 83 28)(11 105 84 27)(12 104 71 26)(13 103 72 25)(14 102 73 24)(29 69 56 98)(30 68 43 97)(31 67 44 96)(32 66 45 95)(33 65 46 94)(34 64 47 93)(35 63 48 92)(36 62 49 91)(37 61 50 90)(38 60 51 89)(39 59 52 88)(40 58 53 87)(41 57 54 86)(42 70 55 85)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(29 70)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 85)(71 112)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)```

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 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 4 4 28 28 28 28 2 2 2 2 4 4 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D7 D14 D14 D14 C7⋊D4 2+ 1+4 D4⋊8D14 kernel C14.1462+ 1+4 C23.21D14 C28⋊7D4 C23⋊D14 C28⋊D4 C28.23D4 C22×D28 C14×C4○D4 C2×C28 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C2×C4 C14 C2 # reps 1 1 4 4 2 2 1 1 4 3 9 9 3 24 2 12

Matrix representation of C14.1462+ 1+4 in GL8(𝔽29)

 26 21 0 0 0 0 0 0 8 21 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 13 13 0 0 0 0 0 0 25 16 0 0 0 0 0 0 0 0 13 0 0 13 0 0 0 0 4 0 1 13 0 0 0 0 2 28 0 1 0 0 0 0 7 0 0 16
,
 21 26 0 0 0 0 0 0 21 8 0 0 0 0 0 0 0 0 13 13 0 0 0 0 0 0 7 16 0 0 0 0 0 0 0 0 13 0 0 13 0 0 0 0 4 0 1 13 0 0 0 0 0 1 0 1 0 0 0 0 25 0 0 16
,
 21 26 0 0 0 0 0 0 21 8 0 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 22 13 0 0 0 0 0 0 0 0 16 13 0 0 0 0 0 0 7 13 0 0 0 0 0 0 27 1 0 28 0 0 0 0 4 16 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 16 0 0 0 0 0 0 4 13 0 0 0 0 0 0 0 0 16 13 0 0 0 0 0 0 25 13 0 0 0 0 0 0 0 1 0 1 0 0 0 0 4 16 1 0

`G:=sub<GL(8,GF(29))| [26,8,0,0,0,0,0,0,21,21,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,13,25,0,0,0,0,0,0,13,16,0,0,0,0,0,0,0,0,13,4,2,7,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,13,13,1,16],[21,21,0,0,0,0,0,0,26,8,0,0,0,0,0,0,0,0,13,7,0,0,0,0,0,0,13,16,0,0,0,0,0,0,0,0,13,4,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,13,1,16],[21,21,0,0,0,0,0,0,26,8,0,0,0,0,0,0,0,0,16,22,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,16,7,27,4,0,0,0,0,13,13,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,4,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,16,25,0,4,0,0,0,0,13,13,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C14.1462+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{146}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.146ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1283);`
`// by ID`

`G=gap.SmallGroup(448,1283);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,184,675,570,18822]);`
`// Polycyclic`

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

׿
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