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## G = C24.2D14order 448 = 26·7

### 2nd non-split extension by C24 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24.2D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×C14 — C2×C23.D7 — C24.2D14
 Lower central C7 — C14 — C2×C14 — C24.2D14
 Upper central C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C24.2D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=abcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde13 >

Subgroups: 628 in 142 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C23, C23, C14, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C2×Dic7, C2×C28, C22×C14, C22×C14, C23.9D4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×C28, C23×C14, C2×C23.D7, C14×C22⋊C4, C24.2D14
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, C23.1D14, C14.C42, C24.2D14

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

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

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

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

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

Matrix representation of C24.2D14 in GL6(𝔽29)

 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 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 18 27 0 0 0 0 2 11 0 0 0 0 0 0 18 27 0 0 0 0 2 11
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 18 27 0 0 0 0 2 11 0 0 0 0 12 0 11 2 0 0 0 12 27 18
,
 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
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 26 3 8 8 0 0 26 22 21 7 0 0 13 20 3 26 0 0 9 25 3 7
,
 17 0 0 0 0 0 0 12 0 0 0 0 0 0 25 25 0 0 0 0 11 4 0 0 0 0 11 8 7 22 0 0 3 18 3 22

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,18,2,0,0,0,0,27,11,0,0,0,0,0,0,18,2,0,0,0,0,27,11],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,18,2,12,0,0,0,27,11,0,12,0,0,0,0,11,27,0,0,0,0,2,18],[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],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,26,26,13,9,0,0,3,22,20,25,0,0,8,21,3,3,0,0,8,7,26,7],[17,0,0,0,0,0,0,12,0,0,0,0,0,0,25,11,11,3,0,0,25,4,8,18,0,0,0,0,7,3,0,0,0,0,22,22] >;`

C24.2D14 in GAP, Magma, Sage, TeX

`C_2^4._2D_{14}`
`% in TeX`

`G:=Group("C2^4.2D14");`
`// GroupNames label`

`G:=SmallGroup(448,84);`
`// by ID`

`G=gap.SmallGroup(448,84);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,1123,851,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=a*b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^13>;`
`// generators/relations`

׿
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