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## G = C24.D14order 448 = 26·7

### 1st non-split extension by C24 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24.D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×C14 — C2×C23.D7 — C24.D14
 Lower central C7 — C14 — C2×C14 — C24.D14
 Upper central C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C24.D14
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

Subgroups: 628 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C2×Dic7, C2×C28, C22×C14, C22×C14, C22×C14, C23.9D4, C23.D7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×C14, C2×C23.D7, C14×C22⋊C4, C24.D14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C23⋊C4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C23.9D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, C23⋊Dic7, C24.D14

Smallest permutation representation of C24.D14
On 112 points
Generators in S112
```(1 102)(2 74)(3 104)(4 76)(5 106)(6 78)(7 108)(8 80)(9 110)(10 82)(11 112)(12 84)(13 86)(14 58)(15 88)(16 60)(17 90)(18 62)(19 92)(20 64)(21 94)(22 66)(23 96)(24 68)(25 98)(26 70)(27 100)(28 72)(29 95)(30 67)(31 97)(32 69)(33 99)(34 71)(35 101)(36 73)(37 103)(38 75)(39 105)(40 77)(41 107)(42 79)(43 109)(44 81)(45 111)(46 83)(47 85)(48 57)(49 87)(50 59)(51 89)(52 61)(53 91)(54 63)(55 93)(56 65)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 65)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(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 85)
(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)
(1 108 36 79)(2 6)(3 106 38 77)(5 104 40 75)(7 102 42 73)(8 28)(9 100 44 71)(10 26)(11 98 46 69)(12 24)(13 96 48 67)(14 22)(15 94 50 65)(16 20)(17 92 52 63)(19 90 54 61)(21 88 56 59)(23 86 30 57)(25 112 32 83)(27 110 34 81)(29 49)(31 47)(33 45)(35 43)(37 41)(51 55)(58 95)(60 93)(62 91)(64 89)(66 87)(68 85)(70 111)(72 109)(74 107)(76 105)(78 103)(80 101)(82 99)(84 97)```

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4L 7A 7B 7C 14A ··· 14U 14V ··· 14AG 28A ··· 28X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + - + - + + image C1 C2 C2 C4 C4 D4 Q8 D7 Dic7 D14 Dic14 C4×D7 D28 C7⋊D4 C23⋊C4 C23⋊Dic7 kernel C24.D14 C2×C23.D7 C14×C22⋊C4 C23.D7 C22×C28 C22×C14 C22×C14 C2×C22⋊C4 C22×C4 C24 C23 C23 C23 C23 C14 C2 # reps 1 2 1 8 4 3 1 3 6 3 6 12 6 12 2 12

Matrix representation of C24.D14 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 20 15 0 0 0 0 14 9 0 0 0 0 0 0 9 14 0 0 0 0 15 20
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 20 15 0 0 0 0 14 9 0 0 0 0 0 0 20 15 0 0 0 0 14 9
,
 1 0 0 0 0 0 0 1 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
,
 12 16 0 0 0 0 0 17 0 0 0 0 0 0 0 0 19 22 0 0 0 0 7 28 0 0 19 22 0 0 0 0 7 28 0 0
,
 12 0 0 0 0 0 11 17 0 0 0 0 0 0 8 10 0 0 0 0 8 21 0 0 0 0 0 0 19 22 0 0 0 0 10 10

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,9,15,0,0,0,0,14,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,20,14,0,0,0,0,15,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,19,7,0,0,0,0,22,28,0,0,19,7,0,0,0,0,22,28,0,0],[12,11,0,0,0,0,0,17,0,0,0,0,0,0,8,8,0,0,0,0,10,21,0,0,0,0,0,0,19,10,0,0,0,0,22,10] >;`

C24.D14 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(448,83);`
`// by ID`

`G=gap.SmallGroup(448,83);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,1684,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;`
`// generators/relations`

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