metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊2D5, C40⋊10D6, C24⋊2D10, D12⋊8D10, C120⋊11C22, D10.13D12, C60.119C23, Dic5.15D12, D60.31C22, Dic30⋊17C22, C8⋊2(S3×D5), C5⋊2C8⋊2D6, C6.4(D4×D5), (C5×D24)⋊4C2, C8⋊D5⋊3S3, C5⋊2(C8⋊D6), (C6×D5).2D4, (C4×D5).2D6, C30.4(C2×D4), C2.9(D5×D12), C24⋊D5⋊9C2, (D5×D12)⋊10C2, C3⋊1(D8⋊D5), C5⋊D24⋊11C2, C15⋊2(C8⋊C22), C10.4(C2×D12), D12⋊5D5⋊8C2, (C3×Dic5).2D4, D12.D5⋊10C2, (C5×D12)⋊15C22, C20.67(C22×S3), (D5×C12).25C22, C12.142(C22×D5), C4.67(C2×S3×D5), (C3×C8⋊D5)⋊3C2, (C3×C5⋊2C8)⋊16C22, SmallGroup(480,326)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊D5
G = < a,b,c,d | a24=b2=c5=d2=1, bab=a-1, ac=ca, dad=a13, bc=cb, dbd=a12b, dcd=c-1 >
Subgroups: 988 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C24, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C22×D5, C24⋊C2, D24, D24, C3×M4(2), C2×D12, C4○D12, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, C8⋊D6, C3×C5⋊2C8, C120, S3×Dic5, C15⋊D4, C5⋊D12, D5×C12, C5×D12, Dic30, D60, C2×S3×D5, D8⋊D5, C5⋊D24, D12.D5, C3×C8⋊D5, C5×D24, C24⋊D5, D12⋊5D5, D5×D12, D24⋊D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C8⋊C22, C22×D5, C2×D12, S3×D5, D4×D5, C8⋊D6, C2×S3×D5, D8⋊D5, D5×D12, D24⋊D5
►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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 24)(20 23)(21 22)(25 26)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(49 52)(50 51)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(73 80)(74 79)(75 78)(76 77)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 102)(98 101)(99 100)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 92 115 66 41)(2 93 116 67 42)(3 94 117 68 43)(4 95 118 69 44)(5 96 119 70 45)(6 73 120 71 46)(7 74 97 72 47)(8 75 98 49 48)(9 76 99 50 25)(10 77 100 51 26)(11 78 101 52 27)(12 79 102 53 28)(13 80 103 54 29)(14 81 104 55 30)(15 82 105 56 31)(16 83 106 57 32)(17 84 107 58 33)(18 85 108 59 34)(19 86 109 60 35)(20 87 110 61 36)(21 88 111 62 37)(22 89 112 63 38)(23 90 113 64 39)(24 91 114 65 40)
(1 41)(2 30)(3 43)(4 32)(5 45)(6 34)(7 47)(8 36)(9 25)(10 38)(11 27)(12 40)(13 29)(14 42)(15 31)(16 44)(17 33)(18 46)(19 35)(20 48)(21 37)(22 26)(23 39)(24 28)(49 87)(50 76)(51 89)(52 78)(53 91)(54 80)(55 93)(56 82)(57 95)(58 84)(59 73)(60 86)(61 75)(62 88)(63 77)(64 90)(65 79)(66 92)(67 81)(68 94)(69 83)(70 96)(71 85)(72 74)(98 110)(100 112)(102 114)(104 116)(106 118)(108 120)
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,24)(20,23)(21,22)(25,26)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(49,52)(50,51)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(73,80)(74,79)(75,78)(76,77)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,102)(98,101)(99,100)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,92,115,66,41)(2,93,116,67,42)(3,94,117,68,43)(4,95,118,69,44)(5,96,119,70,45)(6,73,120,71,46)(7,74,97,72,47)(8,75,98,49,48)(9,76,99,50,25)(10,77,100,51,26)(11,78,101,52,27)(12,79,102,53,28)(13,80,103,54,29)(14,81,104,55,30)(15,82,105,56,31)(16,83,106,57,32)(17,84,107,58,33)(18,85,108,59,34)(19,86,109,60,35)(20,87,110,61,36)(21,88,111,62,37)(22,89,112,63,38)(23,90,113,64,39)(24,91,114,65,40), (1,41)(2,30)(3,43)(4,32)(5,45)(6,34)(7,47)(8,36)(9,25)(10,38)(11,27)(12,40)(13,29)(14,42)(15,31)(16,44)(17,33)(18,46)(19,35)(20,48)(21,37)(22,26)(23,39)(24,28)(49,87)(50,76)(51,89)(52,78)(53,91)(54,80)(55,93)(56,82)(57,95)(58,84)(59,73)(60,86)(61,75)(62,88)(63,77)(64,90)(65,79)(66,92)(67,81)(68,94)(69,83)(70,96)(71,85)(72,74)(98,110)(100,112)(102,114)(104,116)(106,118)(108,120)>;
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,24)(20,23)(21,22)(25,26)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(49,52)(50,51)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(73,80)(74,79)(75,78)(76,77)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,102)(98,101)(99,100)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,92,115,66,41)(2,93,116,67,42)(3,94,117,68,43)(4,95,118,69,44)(5,96,119,70,45)(6,73,120,71,46)(7,74,97,72,47)(8,75,98,49,48)(9,76,99,50,25)(10,77,100,51,26)(11,78,101,52,27)(12,79,102,53,28)(13,80,103,54,29)(14,81,104,55,30)(15,82,105,56,31)(16,83,106,57,32)(17,84,107,58,33)(18,85,108,59,34)(19,86,109,60,35)(20,87,110,61,36)(21,88,111,62,37)(22,89,112,63,38)(23,90,113,64,39)(24,91,114,65,40), (1,41)(2,30)(3,43)(4,32)(5,45)(6,34)(7,47)(8,36)(9,25)(10,38)(11,27)(12,40)(13,29)(14,42)(15,31)(16,44)(17,33)(18,46)(19,35)(20,48)(21,37)(22,26)(23,39)(24,28)(49,87)(50,76)(51,89)(52,78)(53,91)(54,80)(55,93)(56,82)(57,95)(58,84)(59,73)(60,86)(61,75)(62,88)(63,77)(64,90)(65,79)(66,92)(67,81)(68,94)(69,83)(70,96)(71,85)(72,74)(98,110)(100,112)(102,114)(104,116)(106,118)(108,120) );
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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,24),(20,23),(21,22),(25,26),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(37,38),(49,52),(50,51),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(73,80),(74,79),(75,78),(76,77),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(97,102),(98,101),(99,100),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,92,115,66,41),(2,93,116,67,42),(3,94,117,68,43),(4,95,118,69,44),(5,96,119,70,45),(6,73,120,71,46),(7,74,97,72,47),(8,75,98,49,48),(9,76,99,50,25),(10,77,100,51,26),(11,78,101,52,27),(12,79,102,53,28),(13,80,103,54,29),(14,81,104,55,30),(15,82,105,56,31),(16,83,106,57,32),(17,84,107,58,33),(18,85,108,59,34),(19,86,109,60,35),(20,87,110,61,36),(21,88,111,62,37),(22,89,112,63,38),(23,90,113,64,39),(24,91,114,65,40)], [(1,41),(2,30),(3,43),(4,32),(5,45),(6,34),(7,47),(8,36),(9,25),(10,38),(11,27),(12,40),(13,29),(14,42),(15,31),(16,44),(17,33),(18,46),(19,35),(20,48),(21,37),(22,26),(23,39),(24,28),(49,87),(50,76),(51,89),(52,78),(53,91),(54,80),(55,93),(56,82),(57,95),(58,84),(59,73),(60,86),(61,75),(62,88),(63,77),(64,90),(65,79),(66,92),(67,81),(68,94),(69,83),(70,96),(71,85),(72,74),(98,110),(100,112),(102,114),(104,116),(106,118),(108,120)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 12 | 12 | 60 | 2 | 2 | 10 | 60 | 2 | 2 | 2 | 20 | 4 | 20 | 2 | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D12 | D12 | C8⋊C22 | S3×D5 | D4×D5 | C8⋊D6 | C2×S3×D5 | D8⋊D5 | D5×D12 | D24⋊D5 |
| kernel | D24⋊D5 | C5⋊D24 | D12.D5 | C3×C8⋊D5 | C5×D24 | C24⋊D5 | D12⋊5D5 | D5×D12 | C8⋊D5 | C3×Dic5 | C6×D5 | D24 | C5⋊2C8 | C40 | C4×D5 | C24 | D12 | Dic5 | D10 | C15 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D24⋊D5 ►in GL4(𝔽241) generated by
| 2 | 204 | 186 | 174 |
| 37 | 239 | 67 | 55 |
| 55 | 67 | 57 | 30 |
| 174 | 186 | 211 | 184 |
| 239 | 37 | 184 | 211 |
| 204 | 2 | 30 | 57 |
| 186 | 174 | 2 | 204 |
| 67 | 55 | 37 | 239 |
| 0 | 1 | 0 | 0 |
| 240 | 189 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 189 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [2,37,55,174,204,239,67,186,186,67,57,211,174,55,30,184],[239,204,186,67,37,2,174,55,184,30,2,37,211,57,204,239],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
D24⋊D5 in GAP, Magma, Sage, TeX
D_{24}\rtimes D_5 % in TeX
G:=Group("D24:D5"); // GroupNames label
G:=SmallGroup(480,326);
// by ID
G=gap.SmallGroup(480,326);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,142,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^24=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^13,b*c=c*b,d*b*d=a^12*b,d*c*d=c^-1>;
// generators/relations