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## G = C24.41D14order 448 = 26·7

### 41st non-split extension by C24 of D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24.41D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — C2×C4○D28 — C24.41D14
 Lower central C7 — C2×C14 — C24.41D14
 Upper central C1 — C22 — C22×D4

Generators and relations for C24.41D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 1492 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C41D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22×C14, C22.29C24, C4×Dic7, C4⋊Dic7, C23.D7, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×C14, C23.21D14, C28.17D4, C282D4, C28⋊D4, C24⋊D7, C2×C4○D28, D4×C2×C14, C24.41D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, C7⋊D4, C22×D7, C22.29C24, C2×C7⋊D4, C23×D7, D46D14, C22×C7⋊D4, C24.41D14

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

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 7A 7B 7C 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 2 2 2 2 2 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 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D7 D14 D14 D14 C7⋊D4 2+ 1+4 D4⋊6D14 kernel C24.41D14 C23.21D14 C28.17D4 C28⋊2D4 C28⋊D4 C24⋊D7 C2×C4○D28 D4×C2×C14 C2×C28 C22×D4 C22×C4 C2×D4 C24 C2×C4 C14 C2 # reps 1 1 2 4 2 4 1 1 4 3 3 12 6 24 2 12

Matrix representation of C24.41D14 in GL6(𝔽29)

 1 0 0 0 0 0 0 28 0 0 0 0 0 0 24 21 0 0 0 0 3 5 0 0 0 0 6 16 1 7 0 0 9 18 0 28
,
 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 4 0 1 0 0 0 19 0 0 1
,
 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
,
 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
,
 9 0 0 0 0 0 0 13 0 0 0 0 0 0 16 14 0 0 0 0 7 13 0 0 0 0 3 1 13 4 0 0 5 12 17 16
,
 0 16 0 0 0 0 20 0 0 0 0 0 0 0 11 0 19 17 0 0 21 0 22 22 0 0 21 5 20 24 0 0 12 20 8 27

`G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,24,3,6,9,0,0,21,5,16,18,0,0,0,0,1,0,0,0,0,0,7,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,4,19,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[9,0,0,0,0,0,0,13,0,0,0,0,0,0,16,7,3,5,0,0,14,13,1,12,0,0,0,0,13,17,0,0,0,0,4,16],[0,20,0,0,0,0,16,0,0,0,0,0,0,0,11,21,21,12,0,0,0,0,5,20,0,0,19,22,20,8,0,0,17,22,24,27] >;`

C24.41D14 in GAP, Magma, Sage, TeX

`C_2^4._{41}D_{14}`
`% in TeX`

`G:=Group("C2^4.41D14");`
`// GroupNames label`

`G:=SmallGroup(448,1258);`
`// by ID`

`G=gap.SmallGroup(448,1258);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,675,570,18822]);`
`// Polycyclic`

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

׿
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