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G = C14.422+ 1+4order 448 = 26·7

42nd 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 — C14.422+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C14.422+ 1+4
 Lower central C7 — C2×C14 — C14.422+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C14.422+ 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, bd=db, be=eb, dcd-1=ece=a7c, ede=a7b2d >

Subgroups: 1292 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, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, Dic14, C4×D7, C2×Dic7, 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, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23.D14, D7×C22⋊C4, Dic74D4, Dic7.D4, D14⋊Q8, C4⋊C4⋊D7, C4×C7⋊D4, C23.23D14, C23.18D14, C28.17D4, C23⋊D14, C282D4, Dic7⋊D4, C7×C4⋊D4, C14.422+ 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, C14.422+ 1+4

Smallest permutation representation of C14.422+ 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 62 17 77)(2 63 18 78)(3 64 19 79)(4 65 20 80)(5 66 21 81)(6 67 22 82)(7 68 23 83)(8 69 24 84)(9 70 25 71)(10 57 26 72)(11 58 27 73)(12 59 28 74)(13 60 15 75)(14 61 16 76)(29 87 49 105)(30 88 50 106)(31 89 51 107)(32 90 52 108)(33 91 53 109)(34 92 54 110)(35 93 55 111)(36 94 56 112)(37 95 43 99)(38 96 44 100)(39 97 45 101)(40 98 46 102)(41 85 47 103)(42 86 48 104)
(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 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(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 50 24 37)(2 49 25 36)(3 48 26 35)(4 47 27 34)(5 46 28 33)(6 45 15 32)(7 44 16 31)(8 43 17 30)(9 56 18 29)(10 55 19 42)(11 54 20 41)(12 53 21 40)(13 52 22 39)(14 51 23 38)(57 111 79 86)(58 110 80 85)(59 109 81 98)(60 108 82 97)(61 107 83 96)(62 106 84 95)(63 105 71 94)(64 104 72 93)(65 103 73 92)(66 102 74 91)(67 101 75 90)(68 100 76 89)(69 99 77 88)(70 112 78 87)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 43)(25 44)(26 45)(27 46)(28 47)(57 97)(58 98)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 99)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H ··· 4L 7A 7B 7C 14A ··· 14I 14J ··· 14O 14P ··· 14U 28A ··· 28L 28M ··· 28R order 1 2 2 2 2 2 2 2 2 2 4 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 4 4 4 14 14 28 2 2 4 4 4 14 14 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 1 1 1 1 1 2 2 2 2 2 2 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 kernel C14.422+ 1+4 C23.D14 D7×C22⋊C4 Dic7⋊4D4 Dic7.D4 D14⋊Q8 C4⋊C4⋊D7 C4×C7⋊D4 C23.23D14 C23.18D14 C28.17D4 C23⋊D14 C28⋊2D4 Dic7⋊D4 C7×C4⋊D4 C4⋊D4 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 3 4 6 3 3 9 2 12 6

Matrix representation of C14.422+ 1+4 in GL8(𝔽29)

 19 19 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 19 19 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 19 21 0 0 0 0 0 0 17 28 0 0 0 0 0 0 0 0 21 21 0 0 0 0 0 0 8 26
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 17 0 0 4 0 0 0 0 0 17 25 11 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 0 1 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 2 23 28 0 0 0 0 0 6 0 0 28
,
 0 0 28 0 0 0 0 0 0 0 22 1 0 0 0 0 28 0 0 0 0 0 0 0 22 1 0 0 0 0 0 0 0 0 0 0 14 2 0 0 0 0 0 0 3 15 0 0 0 0 0 0 0 0 24 13 0 0 0 0 0 0 27 5
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 20 13 0 0 0 0 0 0 5 9 0 0 0 0 0 0 0 0 24 13 0 0 0 0 0 0 16 5

`G:=sub<GL(8,GF(29))| [19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,19,17,0,0,0,0,0,0,21,28,0,0,0,0,0,0,0,0,21,8,0,0,0,0,0,0,21,26],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,25,12,0,0,0,0,0,4,11,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,2,6,0,0,0,0,0,1,23,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[0,0,28,22,0,0,0,0,0,0,0,1,0,0,0,0,28,22,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,2,15,0,0,0,0,0,0,0,0,24,27,0,0,0,0,0,0,13,5],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,5,0,0,0,0,0,0,13,9,0,0,0,0,0,0,0,0,24,16,0,0,0,0,0,0,13,5] >;`

C14.422+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{42}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.42ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1066);`
`// by ID`

`G=gap.SmallGroup(448,1066);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,297,18822]);`
`// 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,b*d=d*b,b*e=e*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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