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## G = C24⋊D13order 416 = 25·13

### 1st semidirect product of C24 and D13 acting via D13/C13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C24⋊D13
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C13⋊D4 — C24⋊D13
 Lower central C13 — C2×C26 — C24⋊D13
 Upper central C1 — C22 — C24

Generators and relations for C24⋊D13
G = < a,b,c,d,e,f | a2=b2=c2=d2=e13=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 680 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2 [×3], C2 [×7], C4 [×3], C22, C22 [×6], C22 [×17], C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], C13, C22⋊C4 [×3], C2×D4 [×3], C24, D13, C26 [×3], C26 [×6], C22≀C2, Dic13 [×3], D26 [×3], C2×C26, C2×C26 [×6], C2×C26 [×14], C2×Dic13 [×3], C13⋊D4 [×6], C22×D13, C22×C26 [×3], C22×C26 [×6], C23.D13 [×3], C2×C13⋊D4 [×3], C23×C26, C24⋊D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], D13, C22≀C2, D26 [×3], C13⋊D4 [×6], C22×D13, C2×C13⋊D4 [×3], C24⋊D13

Smallest permutation representation of C24⋊D13
On 104 points
Generators in S104
```(1 23)(2 24)(3 25)(4 26)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 40)(35 41)(36 42)(37 43)(38 44)(39 45)
(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 40)(25 41)(26 42)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)
(1 23)(2 24)(3 25)(4 26)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 40)(35 41)(36 42)(37 43)(38 44)(39 45)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)
(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)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 78)(24 77)(25 76)(26 75)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 96)(48 95)(49 94)(50 93)(51 92)(52 104)```

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 13A ··· 13F 26A ··· 26CL order 1 2 2 2 2 ··· 2 2 4 4 4 13 ··· 13 26 ··· 26 size 1 1 1 1 2 ··· 2 52 52 52 52 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D13 D26 C13⋊D4 kernel C24⋊D13 C23.D13 C2×C13⋊D4 C23×C26 C2×C26 C24 C23 C22 # reps 1 3 3 1 6 6 18 72

Matrix representation of C24⋊D13 in GL4(𝔽53) generated by

 52 0 0 0 52 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 1 52 0 0 0 0 1 41 0 0 0 52
,
 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 52 0 0 0 0 52 0 0 0 0 1 0 0 0 0 1
,
 13 0 0 0 35 49 0 0 0 0 16 17 0 0 0 10
,
 1 51 0 0 0 52 0 0 0 0 37 36 0 0 15 16
`G:=sub<GL(4,GF(53))| [52,52,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,52,0,0,0,0,1,0,0,0,41,52],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[13,35,0,0,0,49,0,0,0,0,16,0,0,0,17,10],[1,0,0,0,51,52,0,0,0,0,37,15,0,0,36,16] >;`

C24⋊D13 in GAP, Magma, Sage, TeX

`C_2^4\rtimes D_{13}`
`% in TeX`

`G:=Group("C2^4:D13");`
`// GroupNames label`

`G:=SmallGroup(416,174);`
`// by ID`

`G=gap.SmallGroup(416,174);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,218,13829]);`
`// Polycyclic`

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

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