direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C24, C40⋊3C6, C120⋊9C2, D10.4C12, C12.56D10, C60.69C22, Dic5.4C12, C5⋊3(C2×C24), C5⋊2C8⋊6C6, C15⋊11(C2×C8), (C6×D5).8C4, (C4×D5).7C6, C2.1(D5×C12), C4.12(C6×D5), C6.15(C4×D5), C30.40(C2×C4), C10.8(C2×C12), C20.13(C2×C6), (D5×C12).14C2, (C3×Dic5).8C4, (C3×C5⋊2C8)⋊13C2, SmallGroup(240,33)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×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 62 37 105 96)(2 63 38 106 73)(3 64 39 107 74)(4 65 40 108 75)(5 66 41 109 76)(6 67 42 110 77)(7 68 43 111 78)(8 69 44 112 79)(9 70 45 113 80)(10 71 46 114 81)(11 72 47 115 82)(12 49 48 116 83)(13 50 25 117 84)(14 51 26 118 85)(15 52 27 119 86)(16 53 28 120 87)(17 54 29 97 88)(18 55 30 98 89)(19 56 31 99 90)(20 57 32 100 91)(21 58 33 101 92)(22 59 34 102 93)(23 60 35 103 94)(24 61 36 104 95)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)
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), (1,62,37,105,96)(2,63,38,106,73)(3,64,39,107,74)(4,65,40,108,75)(5,66,41,109,76)(6,67,42,110,77)(7,68,43,111,78)(8,69,44,112,79)(9,70,45,113,80)(10,71,46,114,81)(11,72,47,115,82)(12,49,48,116,83)(13,50,25,117,84)(14,51,26,118,85)(15,52,27,119,86)(16,53,28,120,87)(17,54,29,97,88)(18,55,30,98,89)(19,56,31,99,90)(20,57,32,100,91)(21,58,33,101,92)(22,59,34,102,93)(23,60,35,103,94)(24,61,36,104,95), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)>;
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), (1,62,37,105,96)(2,63,38,106,73)(3,64,39,107,74)(4,65,40,108,75)(5,66,41,109,76)(6,67,42,110,77)(7,68,43,111,78)(8,69,44,112,79)(9,70,45,113,80)(10,71,46,114,81)(11,72,47,115,82)(12,49,48,116,83)(13,50,25,117,84)(14,51,26,118,85)(15,52,27,119,86)(16,53,28,120,87)(17,54,29,97,88)(18,55,30,98,89)(19,56,31,99,90)(20,57,32,100,91)(21,58,33,101,92)(22,59,34,102,93)(23,60,35,103,94)(24,61,36,104,95), (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)(61,116)(62,117)(63,118)(64,119)(65,120)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103) );
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)], [(1,62,37,105,96),(2,63,38,106,73),(3,64,39,107,74),(4,65,40,108,75),(5,66,41,109,76),(6,67,42,110,77),(7,68,43,111,78),(8,69,44,112,79),(9,70,45,113,80),(10,71,46,114,81),(11,72,47,115,82),(12,49,48,116,83),(13,50,25,117,84),(14,51,26,118,85),(15,52,27,119,86),(16,53,28,120,87),(17,54,29,97,88),(18,55,30,98,89),(19,56,31,99,90),(20,57,32,100,91),(21,58,33,101,92),(22,59,34,102,93),(23,60,35,103,94),(24,61,36,104,95)], [(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115),(61,116),(62,117),(63,118),(64,119),(65,120),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103)]])
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 D24⋊7D5 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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