metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊1F5, C120⋊1C4, D5.1D24, C40⋊1Dic3, D10.9D12, D5.1Dic12, Dic5.9Dic6, C8⋊1(C3⋊F5), C5⋊(C24⋊1C4), C3⋊1(D5.D8), (C3×D5).4D8, (C8×D5).3S3, C15⋊2(C2.D8), C6.2(C4⋊F5), C30.9(C4⋊C4), C60.48(C2×C4), (C4×D5).87D6, (C6×D5).52D4, (D5×C24).5C2, C5⋊2C8⋊6Dic3, C12.48(C2×F5), (C3×D5).4Q16, C60⋊C4.8C2, C20.9(C2×Dic3), C2.5(C60⋊C4), C10.2(C4⋊Dic3), (C3×Dic5).10Q8, (D5×C12).112C22, C4.9(C2×C3⋊F5), (C3×C5⋊2C8)⋊10C4, SmallGroup(480,299)
Series: Derived ►Chief ►Lower central ►Upper central
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, C5⋊2C8, C40, C4×D5, C2×F5, C4⋊Dic3, C2×C24, C3×Dic5, C60, C3⋊F5, C6×D5, C8×D5, C4⋊F5, C24⋊1C4, C3×C5⋊2C8, 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, C24⋊1C4, C2×C3⋊F5, D5.D8, C60⋊C4, 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