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## G = C14.1452+ 1+4order 448 = 26·7

### 54th 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 — C14.1452+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D4×D7 — C14.1452+ 1+4
 Lower central C7 — C2×C14 — C14.1452+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C14.1452+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >

Subgroups: 1716 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, D45D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23×D7, C2×D14⋊C4, C4×C7⋊D4, C23.23D14, C287D4, D4×Dic7, C23⋊D14, Dic7⋊D4, D143Q8, C28.23D4, C2×D4×D7, C14×C4○D4, C14.1452+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C7⋊D4, C22×D7, D45D4, C2×C7⋊D4, C23×D7, D7×C4○D4, D48D14, C22×C7⋊D4, C14.1452+ 1+4

Smallest permutation representation of C14.1452+ 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 75 58 90)(2 76 59 91)(3 77 60 92)(4 78 61 93)(5 79 62 94)(6 80 63 95)(7 81 64 96)(8 82 65 97)(9 83 66 98)(10 84 67 85)(11 71 68 86)(12 72 69 87)(13 73 70 88)(14 74 57 89)(15 55 106 37)(16 56 107 38)(17 43 108 39)(18 44 109 40)(19 45 110 41)(20 46 111 42)(21 47 112 29)(22 48 99 30)(23 49 100 31)(24 50 101 32)(25 51 102 33)(26 52 103 34)(27 53 104 35)(28 54 105 36)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 81)(30 82)(31 83)(32 84)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)
(1 90 58 75)(2 91 59 76)(3 92 60 77)(4 93 61 78)(5 94 62 79)(6 95 63 80)(7 96 64 81)(8 97 65 82)(9 98 66 83)(10 85 67 84)(11 86 68 71)(12 87 69 72)(13 88 70 73)(14 89 57 74)(15 55 106 37)(16 56 107 38)(17 43 108 39)(18 44 109 40)(19 45 110 41)(20 46 111 42)(21 47 112 29)(22 48 99 30)(23 49 100 31)(24 50 101 32)(25 51 102 33)(26 52 103 34)(27 53 104 35)(28 54 105 36)
(1 48 8 55)(2 47 9 54)(3 46 10 53)(4 45 11 52)(5 44 12 51)(6 43 13 50)(7 56 14 49)(15 97 22 90)(16 96 23 89)(17 95 24 88)(18 94 25 87)(19 93 26 86)(20 92 27 85)(21 91 28 98)(29 66 36 59)(30 65 37 58)(31 64 38 57)(32 63 39 70)(33 62 40 69)(34 61 41 68)(35 60 42 67)(71 110 78 103)(72 109 79 102)(73 108 80 101)(74 107 81 100)(75 106 82 99)(76 105 83 112)(77 104 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), (1,75,58,90)(2,76,59,91)(3,77,60,92)(4,78,61,93)(5,79,62,94)(6,80,63,95)(7,81,64,96)(8,82,65,97)(9,83,66,98)(10,84,67,85)(11,71,68,86)(12,72,69,87)(13,73,70,88)(14,74,57,89)(15,55,106,37)(16,56,107,38)(17,43,108,39)(18,44,109,40)(19,45,110,41)(20,46,111,42)(21,47,112,29)(22,48,99,30)(23,49,100,31)(24,50,101,32)(25,51,102,33)(26,52,103,34)(27,53,104,35)(28,54,105,36), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104), (1,90,58,75)(2,91,59,76)(3,92,60,77)(4,93,61,78)(5,94,62,79)(6,95,63,80)(7,96,64,81)(8,97,65,82)(9,98,66,83)(10,85,67,84)(11,86,68,71)(12,87,69,72)(13,88,70,73)(14,89,57,74)(15,55,106,37)(16,56,107,38)(17,43,108,39)(18,44,109,40)(19,45,110,41)(20,46,111,42)(21,47,112,29)(22,48,99,30)(23,49,100,31)(24,50,101,32)(25,51,102,33)(26,52,103,34)(27,53,104,35)(28,54,105,36), (1,48,8,55)(2,47,9,54)(3,46,10,53)(4,45,11,52)(5,44,12,51)(6,43,13,50)(7,56,14,49)(15,97,22,90)(16,96,23,89)(17,95,24,88)(18,94,25,87)(19,93,26,86)(20,92,27,85)(21,91,28,98)(29,66,36,59)(30,65,37,58)(31,64,38,57)(32,63,39,70)(33,62,40,69)(34,61,41,68)(35,60,42,67)(71,110,78,103)(72,109,79,102)(73,108,80,101)(74,107,81,100)(75,106,82,99)(76,105,83,112)(77,104,84,111)>;`

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

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

85 conjugacy classes

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

85 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 D14 C7⋊D4 2+ 1+4 D7×C4○D4 D4⋊8D14 kernel C14.1452+ 1+4 C2×D14⋊C4 C4×C7⋊D4 C23.23D14 C28⋊7D4 D4×Dic7 C23⋊D14 Dic7⋊D4 D14⋊3Q8 C28.23D4 C2×D4×D7 C14×C4○D4 C7×D4 C2×C4○D4 D14 C22×C4 C2×D4 C2×Q8 D4 C14 C2 C2 # reps 1 2 1 2 1 1 2 2 1 1 1 1 4 3 4 9 9 3 24 1 6 6

Matrix representation of C14.1452+ 1+4 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 1 8 0 0 12 10
,
 12 0 0 0 0 17 0 0 0 0 20 16 0 0 24 9
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 17 0 0 0 0 17 0 0 0 0 20 16 0 0 24 9
,
 0 12 0 0 12 0 0 0 0 0 9 1 0 0 5 20
`G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,12,0,0,8,10],[12,0,0,0,0,17,0,0,0,0,20,24,0,0,16,9],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[17,0,0,0,0,17,0,0,0,0,20,24,0,0,16,9],[0,12,0,0,12,0,0,0,0,0,9,5,0,0,1,20] >;`

C14.1452+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{145}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.145ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1282);`
`// by ID`

`G=gap.SmallGroup(448,1282);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,136,18822]);`
`// Polycyclic`

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

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