direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C24⋊C2, C40⋊6S3, C24⋊2C10, C120⋊8C2, C15⋊9SD16, C30.28D4, C20.52D6, Dic6⋊1C10, D12.1C10, C10.13D12, C60.64C22, C8⋊2(C5×S3), C6.1(C5×D4), C3⋊1(C5×SD16), C4.8(S3×C10), C2.3(C5×D12), C12.8(C2×C10), (C5×Dic6)⋊7C2, (C5×D12).3C2, SmallGroup(240,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C24⋊C2
G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b11 >
Smallest permutation representation of C5×C24⋊C2
►On 120 points
Generators in S120
(1 86 58 44 109)(2 87 59 45 110)(3 88 60 46 111)(4 89 61 47 112)(5 90 62 48 113)(6 91 63 25 114)(7 92 64 26 115)(8 93 65 27 116)(9 94 66 28 117)(10 95 67 29 118)(11 96 68 30 119)(12 73 69 31 120)(13 74 70 32 97)(14 75 71 33 98)(15 76 72 34 99)(16 77 49 35 100)(17 78 50 36 101)(18 79 51 37 102)(19 80 52 38 103)(20 81 53 39 104)(21 82 54 40 105)(22 83 55 41 106)(23 84 56 42 107)(24 85 57 43 108)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 27)(26 38)(28 36)(29 47)(30 34)(31 45)(33 43)(35 41)(37 39)(40 48)(42 46)(49 55)(50 66)(51 53)(52 64)(54 62)(56 60)(57 71)(59 69)(61 67)(63 65)(68 72)(73 87)(75 85)(76 96)(77 83)(78 94)(79 81)(80 92)(82 90)(84 88)(89 95)(91 93)(98 108)(99 119)(100 106)(101 117)(102 104)(103 115)(105 113)(107 111)(110 120)(112 118)(114 116)
G:=sub<Sym(120)| (1,86,58,44,109)(2,87,59,45,110)(3,88,60,46,111)(4,89,61,47,112)(5,90,62,48,113)(6,91,63,25,114)(7,92,64,26,115)(8,93,65,27,116)(9,94,66,28,117)(10,95,67,29,118)(11,96,68,30,119)(12,73,69,31,120)(13,74,70,32,97)(14,75,71,33,98)(15,76,72,34,99)(16,77,49,35,100)(17,78,50,36,101)(18,79,51,37,102)(19,80,52,38,103)(20,81,53,39,104)(21,82,54,40,105)(22,83,55,41,106)(23,84,56,42,107)(24,85,57,43,108), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,27)(26,38)(28,36)(29,47)(30,34)(31,45)(33,43)(35,41)(37,39)(40,48)(42,46)(49,55)(50,66)(51,53)(52,64)(54,62)(56,60)(57,71)(59,69)(61,67)(63,65)(68,72)(73,87)(75,85)(76,96)(77,83)(78,94)(79,81)(80,92)(82,90)(84,88)(89,95)(91,93)(98,108)(99,119)(100,106)(101,117)(102,104)(103,115)(105,113)(107,111)(110,120)(112,118)(114,116)>;
G:=Group( (1,86,58,44,109)(2,87,59,45,110)(3,88,60,46,111)(4,89,61,47,112)(5,90,62,48,113)(6,91,63,25,114)(7,92,64,26,115)(8,93,65,27,116)(9,94,66,28,117)(10,95,67,29,118)(11,96,68,30,119)(12,73,69,31,120)(13,74,70,32,97)(14,75,71,33,98)(15,76,72,34,99)(16,77,49,35,100)(17,78,50,36,101)(18,79,51,37,102)(19,80,52,38,103)(20,81,53,39,104)(21,82,54,40,105)(22,83,55,41,106)(23,84,56,42,107)(24,85,57,43,108), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,27)(26,38)(28,36)(29,47)(30,34)(31,45)(33,43)(35,41)(37,39)(40,48)(42,46)(49,55)(50,66)(51,53)(52,64)(54,62)(56,60)(57,71)(59,69)(61,67)(63,65)(68,72)(73,87)(75,85)(76,96)(77,83)(78,94)(79,81)(80,92)(82,90)(84,88)(89,95)(91,93)(98,108)(99,119)(100,106)(101,117)(102,104)(103,115)(105,113)(107,111)(110,120)(112,118)(114,116) );
G=PermutationGroup([[(1,86,58,44,109),(2,87,59,45,110),(3,88,60,46,111),(4,89,61,47,112),(5,90,62,48,113),(6,91,63,25,114),(7,92,64,26,115),(8,93,65,27,116),(9,94,66,28,117),(10,95,67,29,118),(11,96,68,30,119),(12,73,69,31,120),(13,74,70,32,97),(14,75,71,33,98),(15,76,72,34,99),(16,77,49,35,100),(17,78,50,36,101),(18,79,51,37,102),(19,80,52,38,103),(20,81,53,39,104),(21,82,54,40,105),(22,83,55,41,106),(23,84,56,42,107),(24,85,57,43,108)], [(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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,27),(26,38),(28,36),(29,47),(30,34),(31,45),(33,43),(35,41),(37,39),(40,48),(42,46),(49,55),(50,66),(51,53),(52,64),(54,62),(56,60),(57,71),(59,69),(61,67),(63,65),(68,72),(73,87),(75,85),(76,96),(77,83),(78,94),(79,81),(80,92),(82,90),(84,88),(89,95),(91,93),(98,108),(99,119),(100,106),(101,117),(102,104),(103,115),(105,113),(107,111),(110,120),(112,118),(114,116)]])
C5×C24⋊C2 is a maximal subgroup of
C24⋊D10 C40⋊14D6 C40⋊8D6 Dic60⋊C2 C40.31D6 Dic6.D10 D30.4D4 C5×S3×SD16
75 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 2 | 2 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | SD16 | D12 | C5×S3 | C5×D4 | C24⋊C2 | S3×C10 | C5×SD16 | C5×D12 | C5×C24⋊C2 |
| kernel | C5×C24⋊C2 | C120 | C5×Dic6 | C5×D12 | C24⋊C2 | C24 | Dic6 | D12 | C40 | C30 | C20 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×C24⋊C2 ►in GL2(𝔽11) generated by
| 5 | 0 |
| 0 | 5 |
| 8 | 9 |
| 1 | 1 |
| 7 | 4 |
| 10 | 4 |
G:=sub<GL(2,GF(11))| [5,0,0,5],[8,1,9,1],[7,10,4,4] >;
C5×C24⋊C2 in GAP, Magma, Sage, TeX
C_5\times C_{24}\rtimes C_2 % in TeX
G:=Group("C5xC24:C2"); // GroupNames label
G:=SmallGroup(240,51);
// by ID
G=gap.SmallGroup(240,51);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,265,127,1443,69,5765]);
// 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^11>;
// generators/relations
Export