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G = C14.1222+ 1+4order 448 = 26·7

31st 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 — C14.1222+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D14⋊C4 — C14.1222+ 1+4
 Lower central C7 — C2×C14 — C14.1222+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.1222+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, bab-1=eae-1=a-1, ac=ca, ad=da, cbc=b-1, dbd-1=a7b, be=eb, dcd-1=a7c, ce=ec, ede-1=a7b2d >

Subgroups: 1228 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, 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×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, 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, C22.45C24, 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, C22⋊Dic14, D7×C22⋊C4, Dic74D4, D14.D4, Dic7.D4, C22.D28, C4⋊C47D7, D142Q8, C4⋊C4⋊D7, C23.21D14, C2×D14⋊C4, D4×Dic7, C23⋊D14, C7×C22.D4, C14.1222+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.45C24, D42D7, C23×D7, C2×D42D7, D7×C4○D4, D48D14, C14.1222+ 1+4

Smallest permutation representation of C14.1222+ 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 71 32 105)(2 84 33 104)(3 83 34 103)(4 82 35 102)(5 81 36 101)(6 80 37 100)(7 79 38 99)(8 78 39 112)(9 77 40 111)(10 76 41 110)(11 75 42 109)(12 74 29 108)(13 73 30 107)(14 72 31 106)(15 52 63 96)(16 51 64 95)(17 50 65 94)(18 49 66 93)(19 48 67 92)(20 47 68 91)(21 46 69 90)(22 45 70 89)(23 44 57 88)(24 43 58 87)(25 56 59 86)(26 55 60 85)(27 54 61 98)(28 53 62 97)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 109)(30 110)(31 111)(32 112)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)
(1 98 32 54)(2 85 33 55)(3 86 34 56)(4 87 35 43)(5 88 36 44)(6 89 37 45)(7 90 38 46)(8 91 39 47)(9 92 40 48)(10 93 41 49)(11 94 42 50)(12 95 29 51)(13 96 30 52)(14 97 31 53)(15 100 63 80)(16 101 64 81)(17 102 65 82)(18 103 66 83)(19 104 67 84)(20 105 68 71)(21 106 69 72)(22 107 70 73)(23 108 57 74)(24 109 58 75)(25 110 59 76)(26 111 60 77)(27 112 61 78)(28 99 62 79)
(1 91 8 98)(2 90 9 97)(3 89 10 96)(4 88 11 95)(5 87 12 94)(6 86 13 93)(7 85 14 92)(15 83 22 76)(16 82 23 75)(17 81 24 74)(18 80 25 73)(19 79 26 72)(20 78 27 71)(21 77 28 84)(29 50 36 43)(30 49 37 56)(31 48 38 55)(32 47 39 54)(33 46 40 53)(34 45 41 52)(35 44 42 51)(57 109 64 102)(58 108 65 101)(59 107 66 100)(60 106 67 99)(61 105 68 112)(62 104 69 111)(63 103 70 110)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 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 7 7 7 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 14 14 28 2 2 4 4 4 4 14 ··· 14 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

67 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 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 C4○D4 D14 D14 D14 D14 2+ 1+4 D4⋊2D7 D7×C4○D4 D4⋊8D14 kernel C14.1222+ 1+4 C22⋊Dic14 D7×C22⋊C4 Dic7⋊4D4 D14.D4 Dic7.D4 C22.D28 C4⋊C4⋊7D7 D14⋊2Q8 C4⋊C4⋊D7 C23.21D14 C2×D14⋊C4 D4×Dic7 C23⋊D14 C7×C22.D4 C22.D4 D14 C2×C14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 3 4 4 9 6 3 3 1 6 6 6

Matrix representation of C14.1222+ 1+4 in GL6(𝔽29)

 26 21 0 0 0 0 8 21 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 8 3 0 0 0 0 8 21 0 0 0 0 0 0 17 24 0 0 0 0 17 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 17 24 0 0 0 0 17 12 0 0 0 0 0 0 0 12 0 0 0 0 17 0
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 2 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 17
,
 8 3 0 0 0 0 8 21 0 0 0 0 0 0 1 27 0 0 0 0 1 28 0 0 0 0 0 0 17 0 0 0 0 0 0 17

`G:=sub<GL(6,GF(29))| [26,8,0,0,0,0,21,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,8,0,0,0,0,3,21,0,0,0,0,0,0,17,17,0,0,0,0,24,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,17,0,0,0,0,24,12,0,0,0,0,0,0,0,17,0,0,0,0,12,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,2,1,0,0,0,0,0,0,12,0,0,0,0,0,0,17],[8,8,0,0,0,0,3,21,0,0,0,0,0,0,1,1,0,0,0,0,27,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;`

C14.1222+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{122}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.122ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1111);`
`// by ID`

`G=gap.SmallGroup(448,1111);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,346,297,136,18822]);`
`// Polycyclic`

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

׿
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