direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D24, C15⋊6D8, C40⋊5S3, C120⋊6C2, C24⋊1C10, D12⋊1C10, C20.53D6, C30.29D4, C10.14D12, C60.65C22, C3⋊1(C5×D8), C8⋊1(C5×S3), C6.2(C5×D4), (C5×D12)⋊7C2, C4.9(S3×C10), C2.4(C5×D12), C12.9(C2×C10), SmallGroup(240,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D24
G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D24
►On 120 points
Generators in S120
(1 90 41 115 56)(2 91 42 116 57)(3 92 43 117 58)(4 93 44 118 59)(5 94 45 119 60)(6 95 46 120 61)(7 96 47 97 62)(8 73 48 98 63)(9 74 25 99 64)(10 75 26 100 65)(11 76 27 101 66)(12 77 28 102 67)(13 78 29 103 68)(14 79 30 104 69)(15 80 31 105 70)(16 81 32 106 71)(17 82 33 107 72)(18 83 34 108 49)(19 84 35 109 50)(20 85 36 110 51)(21 86 37 111 52)(22 87 38 112 53)(23 88 39 113 54)(24 89 40 114 55)
(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 32)(26 31)(27 30)(28 29)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(63 72)(64 71)(65 70)(66 69)(67 68)(73 82)(74 81)(75 80)(76 79)(77 78)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
G:=sub<Sym(120)| (1,90,41,115,56)(2,91,42,116,57)(3,92,43,117,58)(4,93,44,118,59)(5,94,45,119,60)(6,95,46,120,61)(7,96,47,97,62)(8,73,48,98,63)(9,74,25,99,64)(10,75,26,100,65)(11,76,27,101,66)(12,77,28,102,67)(13,78,29,103,68)(14,79,30,104,69)(15,80,31,105,70)(16,81,32,106,71)(17,82,33,107,72)(18,83,34,108,49)(19,84,35,109,50)(20,85,36,110,51)(21,86,37,111,52)(22,87,38,112,53)(23,88,39,113,54)(24,89,40,114,55), (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)(25,32)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(63,72)(64,71)(65,70)(66,69)(67,68)(73,82)(74,81)(75,80)(76,79)(77,78)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)>;
G:=Group( (1,90,41,115,56)(2,91,42,116,57)(3,92,43,117,58)(4,93,44,118,59)(5,94,45,119,60)(6,95,46,120,61)(7,96,47,97,62)(8,73,48,98,63)(9,74,25,99,64)(10,75,26,100,65)(11,76,27,101,66)(12,77,28,102,67)(13,78,29,103,68)(14,79,30,104,69)(15,80,31,105,70)(16,81,32,106,71)(17,82,33,107,72)(18,83,34,108,49)(19,84,35,109,50)(20,85,36,110,51)(21,86,37,111,52)(22,87,38,112,53)(23,88,39,113,54)(24,89,40,114,55), (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)(25,32)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(63,72)(64,71)(65,70)(66,69)(67,68)(73,82)(74,81)(75,80)(76,79)(77,78)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115) );
G=PermutationGroup([[(1,90,41,115,56),(2,91,42,116,57),(3,92,43,117,58),(4,93,44,118,59),(5,94,45,119,60),(6,95,46,120,61),(7,96,47,97,62),(8,73,48,98,63),(9,74,25,99,64),(10,75,26,100,65),(11,76,27,101,66),(12,77,28,102,67),(13,78,29,103,68),(14,79,30,104,69),(15,80,31,105,70),(16,81,32,106,71),(17,82,33,107,72),(18,83,34,108,49),(19,84,35,109,50),(20,85,36,110,51),(21,86,37,111,52),(22,87,38,112,53),(23,88,39,113,54),(24,89,40,114,55)], [(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),(25,32),(26,31),(27,30),(28,29),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(63,72),(64,71),(65,70),(66,69),(67,68),(73,82),(74,81),(75,80),(76,79),(77,78),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(89,90),(97,108),(98,107),(99,106),(100,105),(101,104),(102,103),(109,120),(110,119),(111,118),(112,117),(113,116),(114,115)]])
C5×D24 is a maximal subgroup of
C15⋊D16 C5⋊D48 C15⋊SD32 D24.D5 D24⋊D5 C40⋊5D6 D24⋊6D5 D24⋊7D5 D24⋊5D5 C5×S3×D8
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 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 | 12 | 12 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D4 | D6 | D8 | D12 | C5×S3 | C5×D4 | D24 | S3×C10 | C5×D8 | C5×D12 | C5×D24 |
| kernel | C5×D24 | C120 | C5×D12 | D24 | C24 | D12 | C40 | C30 | C20 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×D24 ►in GL2(𝔽241) generated by
| 91 | 0 |
| 0 | 91 |
| 127 | 105 |
| 136 | 232 |
| 232 | 114 |
| 105 | 9 |
G:=sub<GL(2,GF(241))| [91,0,0,91],[127,136,105,232],[232,105,114,9] >;
C5×D24 in GAP, Magma, Sage, TeX
C_5\times D_{24} % in TeX
G:=Group("C5xD24"); // GroupNames label
G:=SmallGroup(240,52);
// by ID
G=gap.SmallGroup(240,52);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,265,367,1443,69,5765]);
// Polycyclic
G:=Group<a,b,c|a^5=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export