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## G = D5×C24order 240 = 24·3·5

### Direct product of C24 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C24
 Chief series C1 — C5 — C10 — C20 — C60 — D5×C12 — D5×C24
 Lower central C5 — D5×C24
 Upper central C1 — C24

Generators and relations for D5×C24
G = < a,b,c | a24=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C24
On 120 points
Generators in S120
(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 64 28 96 112)(2 65 29 73 113)(3 66 30 74 114)(4 67 31 75 115)(5 68 32 76 116)(6 69 33 77 117)(7 70 34 78 118)(8 71 35 79 119)(9 72 36 80 120)(10 49 37 81 97)(11 50 38 82 98)(12 51 39 83 99)(13 52 40 84 100)(14 53 41 85 101)(15 54 42 86 102)(16 55 43 87 103)(17 56 44 88 104)(18 57 45 89 105)(19 58 46 90 106)(20 59 47 91 107)(21 60 48 92 108)(22 61 25 93 109)(23 62 26 94 110)(24 63 27 95 111)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 97)(23 98)(24 99)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 93)(50 94)(51 95)(52 96)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)

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

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

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

D5×C24 is a maximal subgroup of
C40.51D6  C24.F5  C120.C4  C24⋊F5  C120⋊C4  D5.D24  C40.Dic3  C24.1F5  C40.54D6  C40.34D6  C40.31D6  D247D5  D120⋊C2

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 D5 D10 C3×D5 C4×D5 C6×D5 C8×D5 D5×C12 D5×C24 kernel D5×C24 C3×C5⋊2C8 C120 D5×C12 C8×D5 C3×Dic5 C6×D5 C5⋊2C8 C40 C4×D5 C3×D5 Dic5 D10 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 2 2 4 4 4 8 8 16

Matrix representation of D5×C24 in GL2(𝔽241) generated by

 121 0 0 121
,
 240 1 50 190
,
 1 0 191 240
G:=sub<GL(2,GF(241))| [121,0,0,121],[240,50,1,190],[1,191,0,240] >;

D5×C24 in GAP, Magma, Sage, TeX

D_5\times C_{24}
% in TeX

G:=Group("D5xC24");
// GroupNames label

G:=SmallGroup(240,33);
// by ID

G=gap.SmallGroup(240,33);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,79,69,6917]);
// Polycyclic

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

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