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G = C14.1202+ 1+4order 448 = 26·7

29th 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 — C14.1202+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D4×D7 — C14.1202+ 1+4
 Lower central C7 — C2×C14 — C14.1202+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.1202+ 1+4
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a7b-1, dbd-1=ebe=a7b, cd=dc, ce=ec, ede=a7b2d >

Subgroups: 2220 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22.D4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C233D4, Dic7⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C2×D28, D4×D7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, D7×C22⋊C4, C22⋊D28, C22⋊D28, D14⋊D4, D14.5D4, C4⋊D28, C23.23D14, C23⋊D14, C7×C22.D4, C22×D28, C2×D4×D7, C14.1202+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C22×D7, C233D4, D4×D7, C23×D7, C2×D4×D7, D48D14, C14.1202+ 1+4

Smallest permutation representation of C14.1202+ 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 87 17 111)(2 88 18 112)(3 89 19 99)(4 90 20 100)(5 91 21 101)(6 92 22 102)(7 93 23 103)(8 94 24 104)(9 95 25 105)(10 96 26 106)(11 97 27 107)(12 98 28 108)(13 85 15 109)(14 86 16 110)(29 83 47 60)(30 84 48 61)(31 71 49 62)(32 72 50 63)(33 73 51 64)(34 74 52 65)(35 75 53 66)(36 76 54 67)(37 77 55 68)(38 78 56 69)(39 79 43 70)(40 80 44 57)(41 81 45 58)(42 82 46 59)
(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 71)(70 72)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 99)(97 100)(98 101)
(1 52 24 41)(2 53 25 42)(3 54 26 29)(4 55 27 30)(5 56 28 31)(6 43 15 32)(7 44 16 33)(8 45 17 34)(9 46 18 35)(10 47 19 36)(11 48 20 37)(12 49 21 38)(13 50 22 39)(14 51 23 40)(57 103 73 86)(58 104 74 87)(59 105 75 88)(60 106 76 89)(61 107 77 90)(62 108 78 91)(63 109 79 92)(64 110 80 93)(65 111 81 94)(66 112 82 95)(67 99 83 96)(68 100 84 97)(69 101 71 98)(70 102 72 85)
(1 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 56)(28 55)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 97)(72 96)(73 95)(74 94)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(84 98)```

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A ··· 4E 4F 4G 4H 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 2 2 2 2 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 14 14 28 28 28 4 ··· 4 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

64 irreducible representations

 dim 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 D4 D7 D14 D14 D14 D14 2+ 1+4 D4×D7 D4⋊8D14 kernel C14.1202+ 1+4 D7×C22⋊C4 C22⋊D28 D14⋊D4 D14.5D4 C4⋊D28 C23.23D14 C23⋊D14 C7×C22.D4 C22×D28 C2×D4×D7 C22×D7 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C22 C2 # reps 1 1 3 2 2 2 1 1 1 1 1 4 3 9 6 3 3 2 6 12

Matrix representation of C14.1202+ 1+4 in GL6(𝔽29)

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

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

C14.1202+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{120}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.120ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1106);`
`// by ID`

`G=gap.SmallGroup(448,1106);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,675,297,80,18822]);`
`// 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,a*d=d*a,e*a*e=a^-1,c*b*c=a^7*b^-1,d*b*d^-1=e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;`
`// generators/relations`

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