metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24⋊6D5, C40⋊11D6, C24⋊8D10, D12⋊2D10, D30.1D4, D20.2D6, Dic10⋊2D6, C120⋊19C22, Dic15.1D4, C60.141C23, C8⋊5(S3×D5), (C5×D24)⋊8C2, C40⋊C2⋊3S3, C5⋊2(Q8⋊3D6), C6.30(D4×D5), C40⋊S3⋊7C2, C20⋊D6⋊9C2, C15⋊D8⋊10C2, C3⋊2(D8⋊D5), C15⋊5(C8⋊C22), C10.30(S3×D4), C30.11(C2×D4), D12⋊D5⋊8C2, C15⋊3C8⋊19C22, C20.D6⋊10C2, C2.8(C20⋊D6), (C5×D12)⋊17C22, C20.70(C22×S3), C12.70(C22×D5), (C4×D15).30C22, (C3×D20).26C22, (C3×Dic10)⋊15C22, C4.114(C2×S3×D5), (C3×C40⋊C2)⋊7C2, SmallGroup(480,333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D24⋊6D5
G = < a,b,c,d | a24=b2=c5=d2=1, bab=a-1, ac=ca, dad=a19, bc=cb, dbd=a18b, dcd=c-1 >
Subgroups: 940 in 136 conjugacy classes, 38 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, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, 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, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, 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, Q8⋊3D6, C15⋊3C8, C120, S3×Dic5, C15⋊D4, C5⋊D12, C3×Dic10, C3×D20, C5×D12, C4×D15, C2×S3×D5, D8⋊D5, C15⋊D8, C20.D6, C3×C40⋊C2, C5×D24, C40⋊S3, D12⋊D5, C20⋊D6, D24⋊6D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q8⋊3D6, C2×S3×D5, D8⋊D5, C20⋊D6, D24⋊6D5
►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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(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 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(69 72)(70 71)(73 96)(74 95)(75 94)(76 93)(77 92)(78 91)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(97 104)(98 103)(99 102)(100 101)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
(1 85 101 59 38)(2 86 102 60 39)(3 87 103 61 40)(4 88 104 62 41)(5 89 105 63 42)(6 90 106 64 43)(7 91 107 65 44)(8 92 108 66 45)(9 93 109 67 46)(10 94 110 68 47)(11 95 111 69 48)(12 96 112 70 25)(13 73 113 71 26)(14 74 114 72 27)(15 75 115 49 28)(16 76 116 50 29)(17 77 117 51 30)(18 78 118 52 31)(19 79 119 53 32)(20 80 120 54 33)(21 81 97 55 34)(22 82 98 56 35)(23 83 99 57 36)(24 84 100 58 37)
(1 38)(2 33)(3 28)(4 47)(5 42)(6 37)(7 32)(8 27)(9 46)(10 41)(11 36)(12 31)(13 26)(14 45)(15 40)(16 35)(17 30)(18 25)(19 44)(20 39)(21 34)(22 29)(23 48)(24 43)(49 87)(50 82)(51 77)(52 96)(53 91)(54 86)(55 81)(56 76)(57 95)(58 90)(59 85)(60 80)(61 75)(62 94)(63 89)(64 84)(65 79)(66 74)(67 93)(68 88)(69 83)(70 78)(71 73)(72 92)(98 116)(99 111)(100 106)(102 120)(103 115)(104 110)(107 119)(108 114)(112 118)
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,26)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(69,72)(70,71)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(97,104)(98,103)(99,102)(100,101)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113), (1,85,101,59,38)(2,86,102,60,39)(3,87,103,61,40)(4,88,104,62,41)(5,89,105,63,42)(6,90,106,64,43)(7,91,107,65,44)(8,92,108,66,45)(9,93,109,67,46)(10,94,110,68,47)(11,95,111,69,48)(12,96,112,70,25)(13,73,113,71,26)(14,74,114,72,27)(15,75,115,49,28)(16,76,116,50,29)(17,77,117,51,30)(18,78,118,52,31)(19,79,119,53,32)(20,80,120,54,33)(21,81,97,55,34)(22,82,98,56,35)(23,83,99,57,36)(24,84,100,58,37), (1,38)(2,33)(3,28)(4,47)(5,42)(6,37)(7,32)(8,27)(9,46)(10,41)(11,36)(12,31)(13,26)(14,45)(15,40)(16,35)(17,30)(18,25)(19,44)(20,39)(21,34)(22,29)(23,48)(24,43)(49,87)(50,82)(51,77)(52,96)(53,91)(54,86)(55,81)(56,76)(57,95)(58,90)(59,85)(60,80)(61,75)(62,94)(63,89)(64,84)(65,79)(66,74)(67,93)(68,88)(69,83)(70,78)(71,73)(72,92)(98,116)(99,111)(100,106)(102,120)(103,115)(104,110)(107,119)(108,114)(112,118)>;
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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,26)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(69,72)(70,71)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(97,104)(98,103)(99,102)(100,101)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113), (1,85,101,59,38)(2,86,102,60,39)(3,87,103,61,40)(4,88,104,62,41)(5,89,105,63,42)(6,90,106,64,43)(7,91,107,65,44)(8,92,108,66,45)(9,93,109,67,46)(10,94,110,68,47)(11,95,111,69,48)(12,96,112,70,25)(13,73,113,71,26)(14,74,114,72,27)(15,75,115,49,28)(16,76,116,50,29)(17,77,117,51,30)(18,78,118,52,31)(19,79,119,53,32)(20,80,120,54,33)(21,81,97,55,34)(22,82,98,56,35)(23,83,99,57,36)(24,84,100,58,37), (1,38)(2,33)(3,28)(4,47)(5,42)(6,37)(7,32)(8,27)(9,46)(10,41)(11,36)(12,31)(13,26)(14,45)(15,40)(16,35)(17,30)(18,25)(19,44)(20,39)(21,34)(22,29)(23,48)(24,43)(49,87)(50,82)(51,77)(52,96)(53,91)(54,86)(55,81)(56,76)(57,95)(58,90)(59,85)(60,80)(61,75)(62,94)(63,89)(64,84)(65,79)(66,74)(67,93)(68,88)(69,83)(70,78)(71,73)(72,92)(98,116)(99,111)(100,106)(102,120)(103,115)(104,110)(107,119)(108,114)(112,118) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,26),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(37,38),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(69,72),(70,71),(73,96),(74,95),(75,94),(76,93),(77,92),(78,91),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(97,104),(98,103),(99,102),(100,101),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)], [(1,85,101,59,38),(2,86,102,60,39),(3,87,103,61,40),(4,88,104,62,41),(5,89,105,63,42),(6,90,106,64,43),(7,91,107,65,44),(8,92,108,66,45),(9,93,109,67,46),(10,94,110,68,47),(11,95,111,69,48),(12,96,112,70,25),(13,73,113,71,26),(14,74,114,72,27),(15,75,115,49,28),(16,76,116,50,29),(17,77,117,51,30),(18,78,118,52,31),(19,79,119,53,32),(20,80,120,54,33),(21,81,97,55,34),(22,82,98,56,35),(23,83,99,57,36),(24,84,100,58,37)], [(1,38),(2,33),(3,28),(4,47),(5,42),(6,37),(7,32),(8,27),(9,46),(10,41),(11,36),(12,31),(13,26),(14,45),(15,40),(16,35),(17,30),(18,25),(19,44),(20,39),(21,34),(22,29),(23,48),(24,43),(49,87),(50,82),(51,77),(52,96),(53,91),(54,86),(55,81),(56,76),(57,95),(58,90),(59,85),(60,80),(61,75),(62,94),(63,89),(64,84),(65,79),(66,74),(67,93),(68,88),(69,83),(70,78),(71,73),(72,92),(98,116),(99,111),(100,106),(102,120),(103,115),(104,110),(107,119),(108,114),(112,118)]])
48 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 | 15A | 15B | 20A | 20B | 24A | 24B | 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 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 12 | 20 | 30 | 2 | 2 | 20 | 30 | 2 | 2 | 2 | 40 | 4 | 60 | 2 | 2 | 24 | 24 | 24 | 24 | 4 | 40 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | Q8⋊3D6 | C2×S3×D5 | D8⋊D5 | C20⋊D6 | D24⋊6D5 |
| kernel | D24⋊6D5 | C15⋊D8 | C20.D6 | C3×C40⋊C2 | C5×D24 | C40⋊S3 | D12⋊D5 | C20⋊D6 | C40⋊C2 | Dic15 | D30 | D24 | C40 | Dic10 | D20 | C24 | D12 | C15 | C10 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D24⋊6D5 ►in GL4(𝔽241) generated by
| 232 | 0 | 3 | 184 |
| 0 | 232 | 57 | 229 |
| 238 | 57 | 229 | 57 |
| 184 | 12 | 184 | 3 |
| 232 | 0 | 3 | 184 |
| 0 | 232 | 57 | 229 |
| 12 | 184 | 9 | 0 |
| 57 | 238 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 240 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 51 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [232,0,238,184,0,232,57,12,3,57,229,184,184,229,57,3],[232,0,12,57,0,232,184,238,3,57,9,0,184,229,0,9],[0,240,0,0,1,51,0,0,0,0,0,240,0,0,1,51],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
D24⋊6D5 in GAP, Magma, Sage, TeX
D_{24}\rtimes_6D_5 % in TeX
G:=Group("D24:6D5"); // GroupNames label
G:=SmallGroup(480,333);
// by ID
G=gap.SmallGroup(480,333);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,58,675,346,80,1356,18822]);
// 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^19,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations