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G = C5xC24:C2order 240 = 24·3·5

Direct product of C5 and C24:C2

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5xC24:C2, C40:6S3, C24:2C10, C120:8C2, C15:9SD16, C30.28D4, C20.52D6, Dic6:1C10, D12.1C10, C10.13D12, C60.64C22, C8:2(C5xS3), C6.1(C5xD4), C3:1(C5xSD16), C4.8(S3xC10), C2.3(C5xD12), C12.8(C2xC10), (C5xDic6):7C2, (C5xD12).3C2, SmallGroup(240,51)

Series: Derived Chief Lower central Upper central

C1C12 — C5xC24:C2
C1C3C6C12C60C5xD12 — C5xC24:C2
C3C6C12 — C5xC24:C2
C1C10C20C40

Generators and relations for C5xC24:C2
 G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 104 in 40 conjugacy classes, 22 normal (all characteristic)
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, SD16, C2xC10, D12, C5xS3, C5xD4, C24:C2, S3xC10, C5xSD16, C5xD12, C5xC24:C2
12C2
6C22
6C4
4S3
12C10
3Q8
3D4
2Dic3
2D6
6C20
6C2xC10
4C5xS3
3SD16
3C5xQ8
3C5xD4
2S3xC10
2C5xDic3
3C5xSD16

Smallest permutation representation of C5xC24: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)]])

C5xC24:C2 is a maximal subgroup of
C24:D10  C40:14D6  C40:8D6  Dic60:C2  C40.31D6  Dic6.D10  D30.4D4  C5xS3xSD16

75 conjugacy classes

class 1 2A2B 3 4A4B5A5B5C5D 6 8A8B10A10B10C10D10E10F10G10H12A12B15A15B15C15D20A20B20C20D20E20F20G20H24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order122344555568810101010101010101212151515152020202020202020242424243030303040···4060···60120···120
size111222121111222111112121212222222222212121212222222222···22···22···2

75 irreducible representations

dim11111111222222222222
type++++++++
imageC1C2C2C2C5C10C10C10S3D4D6SD16D12C5xS3C5xD4C24:C2S3xC10C5xSD16C5xD12C5xC24:C2
kernelC5xC24:C2C120C5xDic6C5xD12C24:C2C24Dic6D12C40C30C20C15C10C8C6C5C4C3C2C1
# reps111144441112244448816

Matrix representation of C5xC24:C2 in GL2(F11) generated by

50
05
,
89
11
,
74
104
G:=sub<GL(2,GF(11))| [5,0,0,5],[8,1,9,1],[7,10,4,4] >;

C5xC24: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

Subgroup lattice of C5xC24:C2 in TeX

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