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## G = D5.D24order 480 = 25·3·5

### The non-split extension by D5 of D24 acting via D24/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D5.D24
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×C12 — C60⋊C4 — D5.D24
 Lower central C15 — C30 — C60 — D5.D24
 Upper central C1 — C2 — C4 — C8

Generators and relations for D5.D24
G = < a,b,c,d | a5=b2=c24=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a2, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 460 in 72 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, Dic5, C20, F5, D10, C24, C24, C2×Dic3, C2×C12, C3×D5, C30, C2.D8, C52C8, C40, C4×D5, C2×F5, C4⋊Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, C4⋊F5, C241C4, C3×C52C8, C120, D5×C12, C2×C3⋊F5, D5.D8, D5×C24, C60⋊C4, D5.D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D8, Q16, F5, Dic6, D12, C2×Dic3, C2.D8, C2×F5, D24, Dic12, C4⋊Dic3, C3⋊F5, C4⋊F5, C241C4, C2×C3⋊F5, D5.D8, C60⋊C4, D5.D24

Smallest permutation representation of D5.D24
On 120 points
Generators in S120
```(1 77 118 32 49)(2 78 119 33 50)(3 79 120 34 51)(4 80 97 35 52)(5 81 98 36 53)(6 82 99 37 54)(7 83 100 38 55)(8 84 101 39 56)(9 85 102 40 57)(10 86 103 41 58)(11 87 104 42 59)(12 88 105 43 60)(13 89 106 44 61)(14 90 107 45 62)(15 91 108 46 63)(16 92 109 47 64)(17 93 110 48 65)(18 94 111 25 66)(19 95 112 26 67)(20 96 113 27 68)(21 73 114 28 69)(22 74 115 29 70)(23 75 116 30 71)(24 76 117 31 72)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 94)(26 95)(27 96)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 70 111 74)(26 69 112 73)(27 68 113 96)(28 67 114 95)(29 66 115 94)(30 65 116 93)(31 64 117 92)(32 63 118 91)(33 62 119 90)(34 61 120 89)(35 60 97 88)(36 59 98 87)(37 58 99 86)(38 57 100 85)(39 56 101 84)(40 55 102 83)(41 54 103 82)(42 53 104 81)(43 52 105 80)(44 51 106 79)(45 50 107 78)(46 49 108 77)(47 72 109 76)(48 71 110 75)```

`G:=sub<Sym(120)| (1,77,118,32,49)(2,78,119,33,50)(3,79,120,34,51)(4,80,97,35,52)(5,81,98,36,53)(6,82,99,37,54)(7,83,100,38,55)(8,84,101,39,56)(9,85,102,40,57)(10,86,103,41,58)(11,87,104,42,59)(12,88,105,43,60)(13,89,106,44,61)(14,90,107,45,62)(15,91,108,46,63)(16,92,109,47,64)(17,93,110,48,65)(18,94,111,25,66)(19,95,112,26,67)(20,96,113,27,68)(21,73,114,28,69)(22,74,115,29,70)(23,75,116,30,71)(24,76,117,31,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,94)(26,95)(27,96)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,70,111,74)(26,69,112,73)(27,68,113,96)(28,67,114,95)(29,66,115,94)(30,65,116,93)(31,64,117,92)(32,63,118,91)(33,62,119,90)(34,61,120,89)(35,60,97,88)(36,59,98,87)(37,58,99,86)(38,57,100,85)(39,56,101,84)(40,55,102,83)(41,54,103,82)(42,53,104,81)(43,52,105,80)(44,51,106,79)(45,50,107,78)(46,49,108,77)(47,72,109,76)(48,71,110,75)>;`

`G:=Group( (1,77,118,32,49)(2,78,119,33,50)(3,79,120,34,51)(4,80,97,35,52)(5,81,98,36,53)(6,82,99,37,54)(7,83,100,38,55)(8,84,101,39,56)(9,85,102,40,57)(10,86,103,41,58)(11,87,104,42,59)(12,88,105,43,60)(13,89,106,44,61)(14,90,107,45,62)(15,91,108,46,63)(16,92,109,47,64)(17,93,110,48,65)(18,94,111,25,66)(19,95,112,26,67)(20,96,113,27,68)(21,73,114,28,69)(22,74,115,29,70)(23,75,116,30,71)(24,76,117,31,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,94)(26,95)(27,96)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,70,111,74)(26,69,112,73)(27,68,113,96)(28,67,114,95)(29,66,115,94)(30,65,116,93)(31,64,117,92)(32,63,118,91)(33,62,119,90)(34,61,120,89)(35,60,97,88)(36,59,98,87)(37,58,99,86)(38,57,100,85)(39,56,101,84)(40,55,102,83)(41,54,103,82)(42,53,104,81)(43,52,105,80)(44,51,106,79)(45,50,107,78)(46,49,108,77)(47,72,109,76)(48,71,110,75) );`

`G=PermutationGroup([[(1,77,118,32,49),(2,78,119,33,50),(3,79,120,34,51),(4,80,97,35,52),(5,81,98,36,53),(6,82,99,37,54),(7,83,100,38,55),(8,84,101,39,56),(9,85,102,40,57),(10,86,103,41,58),(11,87,104,42,59),(12,88,105,43,60),(13,89,106,44,61),(14,90,107,45,62),(15,91,108,46,63),(16,92,109,47,64),(17,93,110,48,65),(18,94,111,25,66),(19,95,112,26,67),(20,96,113,27,68),(21,73,114,28,69),(22,74,115,29,70),(23,75,116,30,71),(24,76,117,31,72)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,94),(26,95),(27,96),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(37,82),(38,83),(39,84),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(25,70,111,74),(26,69,112,73),(27,68,113,96),(28,67,114,95),(29,66,115,94),(30,65,116,93),(31,64,117,92),(32,63,118,91),(33,62,119,90),(34,61,120,89),(35,60,97,88),(36,59,98,87),(37,58,99,86),(38,57,100,85),(39,56,101,84),(40,55,102,83),(41,54,103,82),(42,53,104,81),(43,52,105,80),(44,51,106,79),(45,50,107,78),(46,49,108,77),(47,72,109,76),(48,71,110,75)]])`

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 8A 8B 8C 8D 10 12A 12B 12C 12D 15A 15B 20A 20B 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 4 4 4 4 4 4 5 6 6 6 8 8 8 8 10 12 12 12 12 15 15 20 20 24 24 24 24 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 5 5 2 2 10 60 60 60 60 4 2 10 10 2 2 10 10 4 2 2 10 10 4 4 4 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + - + - - + + - - + + - + + image C1 C2 C2 C4 C4 S3 Q8 D4 Dic3 Dic3 D6 D8 Q16 Dic6 D12 D24 Dic12 F5 C2×F5 C3⋊F5 C4⋊F5 C2×C3⋊F5 D5.D8 C60⋊C4 D5.D24 kernel D5.D24 D5×C24 C60⋊C4 C3×C5⋊2C8 C120 C8×D5 C3×Dic5 C6×D5 C5⋊2C8 C40 C4×D5 C3×D5 C3×D5 Dic5 D10 D5 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 1 1 1 1 1 1 2 2 2 2 4 4 1 1 2 2 2 4 4 8

Matrix representation of D5.D24 in GL8(𝔽241)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 240
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 240 240 240 0 0 0 0 0 0 0 1
,
 145 175 0 0 0 0 0 0 104 74 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 224 0 207 207 0 0 0 0 34 17 34 0 0 0 0 0 0 34 17 34 0 0 0 0 207 207 0 224
,
 144 143 0 0 0 0 0 0 37 97 0 0 0 0 0 0 0 0 177 0 0 0 0 0 0 0 177 64 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 1 1 1 1 0 0 0 0 0 240 0 0

`G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,1],[145,104,0,0,0,0,0,0,175,74,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,224,34,0,207,0,0,0,0,0,17,34,207,0,0,0,0,207,34,17,0,0,0,0,0,207,0,34,224],[144,37,0,0,0,0,0,0,143,97,0,0,0,0,0,0,0,0,177,177,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,240,0,1,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,1,0,0,0,0,0,0,0,1,0] >;`

D5.D24 in GAP, Magma, Sage, TeX

`D_5.D_{24}`
`% in TeX`

`G:=Group("D5.D24");`
`// GroupNames label`

`G:=SmallGroup(480,299);`
`// by ID`

`G=gap.SmallGroup(480,299);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,675,80,2693,14118,4724]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^24=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^2,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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