metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊7D6, C24⋊1D10, D120⋊14C2, C120⋊7C22, Dic6⋊8D10, D10.12D12, D12.19D10, D60⋊19C22, C60.118C23, Dic5.14D12, C8⋊1(S3×D5), C5⋊2C8⋊1D6, C6.3(D4×D5), (D5×D12)⋊9C2, C8⋊D5⋊1S3, C24⋊C2⋊1D5, C5⋊1(C8⋊D6), (C6×D5).1D4, (C4×D5).1D6, C30.3(C2×D4), C2.8(D5×D12), C5⋊D24⋊10C2, C3⋊1(D40⋊C2), C15⋊1(C8⋊C22), C10.3(C2×D12), C12.28D10⋊8C2, (C3×Dic5).1D4, Dic6⋊D5⋊10C2, C20.66(C22×S3), (C5×Dic6)⋊13C22, (D5×C12).24C22, (C5×D12).21C22, C12.141(C22×D5), C4.66(C2×S3×D5), (C5×C24⋊C2)⋊1C2, (C3×C8⋊D5)⋊1C2, (C3×C5⋊2C8)⋊15C22, SmallGroup(480,325)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊D10
G = < a,b,c | a24=b10=c2=1, bab-1=a11, cac=a-1, cbc=b-1 >
Subgroups: 1052 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, C20, 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, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C5×Dic3, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, C8⋊D6, C3×C5⋊2C8, C120, D30.C2, C3⋊D20, C5⋊D12, D5×C12, C5×Dic6, C5×D12, D60, C2×S3×D5, D40⋊C2, C5⋊D24, Dic6⋊D5, C3×C8⋊D5, C5×C24⋊C2, D120, C12.28D10, D5×D12, C24⋊D10
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, D40⋊C2, D5×D12, C24⋊D10
►On 120 points
(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)
(1 119 73 64 33 7 113 79 58 39)(2 106 74 51 34 18 114 90 59 26)(3 117 75 62 35 5 115 77 60 37)(4 104 76 49 36 16 116 88 61 48)(6 102 78 71 38 14 118 86 63 46)(8 100 80 69 40 12 120 84 65 44)(9 111 81 56 41 23 97 95 66 31)(10 98 82 67 42)(11 109 83 54 43 21 99 93 68 29)(13 107 85 52 45 19 101 91 70 27)(15 105 87 50 47 17 103 89 72 25)(20 112 92 57 28 24 108 96 53 32)(22 110 94 55 30)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 79)(74 78)(75 77)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)
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), (1,119,73,64,33,7,113,79,58,39)(2,106,74,51,34,18,114,90,59,26)(3,117,75,62,35,5,115,77,60,37)(4,104,76,49,36,16,116,88,61,48)(6,102,78,71,38,14,118,86,63,46)(8,100,80,69,40,12,120,84,65,44)(9,111,81,56,41,23,97,95,66,31)(10,98,82,67,42)(11,109,83,54,43,21,99,93,68,29)(13,107,85,52,45,19,101,91,70,27)(15,105,87,50,47,17,103,89,72,25)(20,112,92,57,28,24,108,96,53,32)(22,110,94,55,30), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,79)(74,78)(75,77)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)>;
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), (1,119,73,64,33,7,113,79,58,39)(2,106,74,51,34,18,114,90,59,26)(3,117,75,62,35,5,115,77,60,37)(4,104,76,49,36,16,116,88,61,48)(6,102,78,71,38,14,118,86,63,46)(8,100,80,69,40,12,120,84,65,44)(9,111,81,56,41,23,97,95,66,31)(10,98,82,67,42)(11,109,83,54,43,21,99,93,68,29)(13,107,85,52,45,19,101,91,70,27)(15,105,87,50,47,17,103,89,72,25)(20,112,92,57,28,24,108,96,53,32)(22,110,94,55,30), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,79)(74,78)(75,77)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89) );
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)], [(1,119,73,64,33,7,113,79,58,39),(2,106,74,51,34,18,114,90,59,26),(3,117,75,62,35,5,115,77,60,37),(4,104,76,49,36,16,116,88,61,48),(6,102,78,71,38,14,118,86,63,46),(8,100,80,69,40,12,120,84,65,44),(9,111,81,56,41,23,97,95,66,31),(10,98,82,67,42),(11,109,83,54,43,21,99,93,68,29),(13,107,85,52,45,19,101,91,70,27),(15,105,87,50,47,17,103,89,72,25),(20,112,92,57,28,24,108,96,53,32),(22,110,94,55,30)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(73,79),(74,78),(75,77),(80,96),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 12 | 60 | 60 | 2 | 2 | 10 | 12 | 2 | 2 | 2 | 20 | 4 | 20 | 2 | 2 | 24 | 24 | 2 | 2 | 20 | 4 | 4 | 4 | 4 | 24 | 24 | 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 | 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 | D10 | D12 | D12 | C8⋊C22 | S3×D5 | D4×D5 | C8⋊D6 | C2×S3×D5 | D40⋊C2 | D5×D12 | C24⋊D10 |
| kernel | C24⋊D10 | C5⋊D24 | Dic6⋊D5 | C3×C8⋊D5 | C5×C24⋊C2 | D120 | C12.28D10 | D5×D12 | C8⋊D5 | C3×Dic5 | C6×D5 | C24⋊C2 | C5⋊2C8 | C40 | C4×D5 | C24 | Dic6 | 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 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C24⋊D10 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 143 | 100 | 2 | 2 |
| 0 | 0 | 141 | 43 | 239 | 0 |
| 0 | 0 | 170 | 71 | 98 | 141 |
| 0 | 0 | 170 | 99 | 100 | 198 |
| 0 | 189 | 0 | 0 | 0 | 0 |
| 51 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 198 | 0 | 0 |
| 0 | 0 | 99 | 142 | 0 | 0 |
| 0 | 0 | 141 | 43 | 240 | 0 |
| 0 | 0 | 143 | 100 | 1 | 1 |
| 51 | 189 | 0 | 0 | 0 | 0 |
| 50 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 198 | 0 | 0 |
| 0 | 0 | 99 | 142 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,143,141,170,170,0,0,100,43,71,99,0,0,2,239,98,100,0,0,2,0,141,198],[0,51,0,0,0,0,189,190,0,0,0,0,0,0,99,99,141,143,0,0,198,142,43,100,0,0,0,0,240,1,0,0,0,0,0,1],[51,50,0,0,0,0,189,190,0,0,0,0,0,0,99,99,0,0,0,0,198,142,0,0,0,0,0,0,1,240,0,0,0,0,0,240] >;
C24⋊D10 in GAP, Magma, Sage, TeX
C_{24}\rtimes D_{10} % in TeX
G:=Group("C24:D10"); // GroupNames label
G:=SmallGroup(480,325);
// by ID
G=gap.SmallGroup(480,325);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,58,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^24=b^10=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations