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## G = C14.402+ 1+4order 448 = 26·7

### 40th 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 — C14.402+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D4×D7 — C14.402+ 1+4
 Lower central C7 — C2×C14 — C14.402+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C14.402+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >

Subgroups: 1836 in 334 conjugacy classes, 105 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, C4○D4, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, D45D4, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, D4×D7, D42D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, D7×C22⋊C4, Dic74D4, D14⋊D4, Dic7.D4, D14.5D4, D14⋊Q8, C2×D14⋊C4, C4×C7⋊D4, C23.18D14, C23⋊D14, Dic7⋊D4, C7×C4⋊D4, C2×D4×D7, C2×D42D7, C14.402+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D45D4, D4×D7, C23×D7, C2×D4×D7, D46D14, D7×C4○D4, C14.402+ 1+4

Smallest permutation representation of C14.402+ 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 82 21 68)(2 83 22 69)(3 84 23 70)(4 71 24 57)(5 72 25 58)(6 73 26 59)(7 74 27 60)(8 75 28 61)(9 76 15 62)(10 77 16 63)(11 78 17 64)(12 79 18 65)(13 80 19 66)(14 81 20 67)(29 105 43 96)(30 106 44 97)(31 107 45 98)(32 108 46 85)(33 109 47 86)(34 110 48 87)(35 111 49 88)(36 112 50 89)(37 99 51 90)(38 100 52 91)(39 101 53 92)(40 102 54 93)(41 103 55 94)(42 104 56 95)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 57)(12 58)(13 59)(14 60)(15 83)(16 84)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 96)(30 97)(31 98)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)
(1 45 21 31)(2 44 22 30)(3 43 23 29)(4 56 24 42)(5 55 25 41)(6 54 26 40)(7 53 27 39)(8 52 28 38)(9 51 15 37)(10 50 16 36)(11 49 17 35)(12 48 18 34)(13 47 19 33)(14 46 20 32)(57 111 71 88)(58 110 72 87)(59 109 73 86)(60 108 74 85)(61 107 75 98)(62 106 76 97)(63 105 77 96)(64 104 78 95)(65 103 79 94)(66 102 80 93)(67 101 81 92)(68 100 82 91)(69 99 83 90)(70 112 84 89)
(1 31 8 38)(2 32 9 39)(3 33 10 40)(4 34 11 41)(5 35 12 42)(6 36 13 29)(7 37 14 30)(15 53 22 46)(16 54 23 47)(17 55 24 48)(18 56 25 49)(19 43 26 50)(20 44 27 51)(21 45 28 52)(57 94 64 87)(58 95 65 88)(59 96 66 89)(60 97 67 90)(61 98 68 91)(62 85 69 92)(63 86 70 93)(71 103 78 110)(72 104 79 111)(73 105 80 112)(74 106 81 99)(75 107 82 100)(76 108 83 101)(77 109 84 102)```

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

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

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

67 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 ··· 14O 14P ··· 14U 28A ··· 28L 28M ··· 28R 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 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 4 14 14 14 14 28 2 2 4 4 4 14 14 14 14 28 28 28 2 2 2 2 ··· 2 4 ··· 4 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 D4 D7 C4○D4 D14 D14 D14 D14 2+ 1+4 D4×D7 D4⋊6D14 D7×C4○D4 kernel C14.402+ 1+4 D7×C22⋊C4 Dic7⋊4D4 D14⋊D4 Dic7.D4 D14.5D4 D14⋊Q8 C2×D14⋊C4 C4×C7⋊D4 C23.18D14 C23⋊D14 Dic7⋊D4 C7×C4⋊D4 C2×D4×D7 C2×D4⋊2D7 C7⋊D4 C4⋊D4 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 4 3 4 6 3 3 9 1 6 6 6

Matrix representation of C14.402+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 10 10 0 0 0 0 19 22 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 18 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 3 17 0 0 0 0 25 26
,
 28 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 26 12 0 0 0 0 9 3
,
 11 2 0 0 0 0 27 18 0 0 0 0 0 0 28 0 0 0 0 0 22 1 0 0 0 0 0 0 7 1 0 0 0 0 8 22
,
 18 27 0 0 0 0 3 11 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 22 28 0 0 0 0 21 7

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,18,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,25,0,0,0,0,17,26],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,9,0,0,0,0,12,3],[11,27,0,0,0,0,2,18,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,1,22],[18,3,0,0,0,0,27,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,21,0,0,0,0,28,7] >;`

C14.402+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{40}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.40ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1063);`
`// by ID`

`G=gap.SmallGroup(448,1063);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,297,136,18822]);`
`// Polycyclic`

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