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## G = D14.C24order 448 = 26·7

### 9th non-split extension by D14 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D14.C24
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7 — D7×C4○D4 — D14.C24
 Lower central C7 — C14 — D14.C24
 Upper central C1 — C2 — 2+ 1+4

Generators and relations for D14.C24
G = < a,b,c,d,e,f | a14=b2=c2=e2=f2=1, d2=a7, bab=eae=a-1, ac=ca, ad=da, af=fa, cbc=fbf=a7b, bd=db, ebe=a12b, cd=dc, ce=ec, cf=fc, de=ed, fdf=a7d, fef=a7e >

Subgroups: 3060 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, C14, C14, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C2.C25, C2×Dic14, C2×C4×D7, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, C22×Dic7, C2×C7⋊D4, D4×C14, C7×C4○D4, C2×D42D7, D46D14, D7×C4○D4, D4.10D14, C7×2+ 1+4, D14.C24
Quotients: C1, C2, C22, C23, D7, C24, D14, C25, C22×D7, C2.C25, C23×D7, D7×C24, D14.C24

Smallest permutation representation of D14.C24
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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 23)(16 22)(17 21)(18 20)(24 28)(25 27)(29 38)(30 37)(31 36)(32 35)(33 34)(39 42)(40 41)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 67)(58 66)(59 65)(60 64)(61 63)(68 70)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 95)(86 94)(87 93)(88 92)(89 91)(96 98)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 71)(28 72)(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)(43 108)(44 109)(45 110)(46 111)(47 112)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)
(1 34 8 41)(2 35 9 42)(3 36 10 29)(4 37 11 30)(5 38 12 31)(6 39 13 32)(7 40 14 33)(15 49 22 56)(16 50 23 43)(17 51 24 44)(18 52 25 45)(19 53 26 46)(20 54 27 47)(21 55 28 48)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 112 78 105)(72 99 79 106)(73 100 80 107)(74 101 81 108)(75 102 82 109)(76 103 83 110)(77 104 84 111)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 24)(16 23)(17 22)(18 21)(19 20)(25 28)(26 27)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(43 50)(44 49)(45 48)(46 47)(51 56)(52 55)(53 54)(57 61)(58 60)(62 70)(63 69)(64 68)(65 67)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 89)(86 88)(90 98)(91 97)(92 96)(93 95)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)(111 112)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 15)(29 52)(30 53)(31 54)(32 55)(33 56)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(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)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)```

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

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

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

85 conjugacy classes

 class 1 2A 2B ··· 2J 2K ··· 2P 4A ··· 4F 4G 4H 4I ··· 4Q 7A 7B 7C 14A 14B 14C 14D ··· 14AD 28A ··· 28R order 1 2 2 ··· 2 2 ··· 2 4 ··· 4 4 4 4 ··· 4 7 7 7 14 14 14 14 ··· 14 28 ··· 28 size 1 1 2 ··· 2 14 ··· 14 2 ··· 2 7 7 14 ··· 14 2 2 2 2 2 2 4 ··· 4 4 ··· 4

85 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D7 D14 D14 C2.C25 D14.C24 kernel D14.C24 C2×D4⋊2D7 D4⋊6D14 D7×C4○D4 D4.10D14 C7×2+ 1+4 2+ 1+4 C2×D4 C4○D4 C7 C1 # reps 1 9 9 6 6 1 3 27 18 2 3

Matrix representation of D14.C24 in GL6(𝔽29)

 25 25 0 0 0 0 4 11 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
,
 25 25 0 0 0 0 11 4 0 0 0 0 0 0 28 0 19 0 0 0 0 1 0 4 0 0 0 0 1 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 6 13 13 0 0 25 26 1 1 0 0 16 0 22 23 0 0 0 16 4 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 0 0 9 0 0 0 12 6 0 0 0 0 0 17 0 0 0 0 0 0 12
,
 28 0 0 0 0 0 11 1 0 0 0 0 0 0 1 0 0 21 0 0 0 28 14 0 0 0 0 0 1 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 26 26 0 0 1 0 3 3 0 0 0 0 0 1 0 0 0 0 1 0

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

D14.C24 in GAP, Magma, Sage, TeX

`D_{14}.C_2^4`
`% in TeX`

`G:=Group("D14.C2^4");`
`// GroupNames label`

`G:=SmallGroup(448,1380);`
`// by ID`

`G=gap.SmallGroup(448,1380);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,1684,438,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^14=b^2=c^2=e^2=f^2=1,d^2=a^7,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,a*f=f*a,c*b*c=f*b*f=a^7*b,b*d=d*b,e*b*e=a^12*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=a^7*d,f*e*f=a^7*e>;`
`// generators/relations`

׿
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