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G = C5×C24⋊C2order 240 = 24·3·5

Direct product of C5 and C24⋊C2

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

Aliases: C5×C24⋊C2, C406S3, C242C10, C1208C2, C159SD16, C30.28D4, C20.52D6, Dic61C10, D12.1C10, C10.13D12, C60.64C22, C82(C5×S3), C6.1(C5×D4), C31(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

C1C12 — C5×C24⋊C2
C1C3C6C12C60C5×D12 — C5×C24⋊C2
C3C6C12 — C5×C24⋊C2
C1C10C20C40

Generators and relations for C5×C24⋊C2
 G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

12C2
6C22
6C4
4S3
12C10
3Q8
3D4
2Dic3
2D6
6C20
6C2×C10
4C5×S3
3SD16
3C5×Q8
3C5×D4
2S3×C10
2C5×Dic3
3C5×SD16

Smallest permutation representation of C5×C24⋊C2
On 120 points
Generators in S120
(1 102 26 69 73)(2 103 27 70 74)(3 104 28 71 75)(4 105 29 72 76)(5 106 30 49 77)(6 107 31 50 78)(7 108 32 51 79)(8 109 33 52 80)(9 110 34 53 81)(10 111 35 54 82)(11 112 36 55 83)(12 113 37 56 84)(13 114 38 57 85)(14 115 39 58 86)(15 116 40 59 87)(16 117 41 60 88)(17 118 42 61 89)(18 119 43 62 90)(19 120 44 63 91)(20 97 45 64 92)(21 98 46 65 93)(22 99 47 66 94)(23 100 48 67 95)(24 101 25 68 96)
(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 39)(27 37)(28 48)(29 35)(30 46)(31 33)(32 44)(34 42)(36 40)(41 47)(43 45)(49 65)(50 52)(51 63)(53 61)(54 72)(55 59)(56 70)(58 68)(60 66)(62 64)(67 71)(74 84)(75 95)(76 82)(77 93)(78 80)(79 91)(81 89)(83 87)(86 96)(88 94)(90 92)(97 119)(98 106)(99 117)(100 104)(101 115)(103 113)(105 111)(107 109)(108 120)(110 118)(112 116)

G:=sub<Sym(120)| (1,102,26,69,73)(2,103,27,70,74)(3,104,28,71,75)(4,105,29,72,76)(5,106,30,49,77)(6,107,31,50,78)(7,108,32,51,79)(8,109,33,52,80)(9,110,34,53,81)(10,111,35,54,82)(11,112,36,55,83)(12,113,37,56,84)(13,114,38,57,85)(14,115,39,58,86)(15,116,40,59,87)(16,117,41,60,88)(17,118,42,61,89)(18,119,43,62,90)(19,120,44,63,91)(20,97,45,64,92)(21,98,46,65,93)(22,99,47,66,94)(23,100,48,67,95)(24,101,25,68,96), (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)(25,39)(27,37)(28,48)(29,35)(30,46)(31,33)(32,44)(34,42)(36,40)(41,47)(43,45)(49,65)(50,52)(51,63)(53,61)(54,72)(55,59)(56,70)(58,68)(60,66)(62,64)(67,71)(74,84)(75,95)(76,82)(77,93)(78,80)(79,91)(81,89)(83,87)(86,96)(88,94)(90,92)(97,119)(98,106)(99,117)(100,104)(101,115)(103,113)(105,111)(107,109)(108,120)(110,118)(112,116)>;

G:=Group( (1,102,26,69,73)(2,103,27,70,74)(3,104,28,71,75)(4,105,29,72,76)(5,106,30,49,77)(6,107,31,50,78)(7,108,32,51,79)(8,109,33,52,80)(9,110,34,53,81)(10,111,35,54,82)(11,112,36,55,83)(12,113,37,56,84)(13,114,38,57,85)(14,115,39,58,86)(15,116,40,59,87)(16,117,41,60,88)(17,118,42,61,89)(18,119,43,62,90)(19,120,44,63,91)(20,97,45,64,92)(21,98,46,65,93)(22,99,47,66,94)(23,100,48,67,95)(24,101,25,68,96), (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)(25,39)(27,37)(28,48)(29,35)(30,46)(31,33)(32,44)(34,42)(36,40)(41,47)(43,45)(49,65)(50,52)(51,63)(53,61)(54,72)(55,59)(56,70)(58,68)(60,66)(62,64)(67,71)(74,84)(75,95)(76,82)(77,93)(78,80)(79,91)(81,89)(83,87)(86,96)(88,94)(90,92)(97,119)(98,106)(99,117)(100,104)(101,115)(103,113)(105,111)(107,109)(108,120)(110,118)(112,116) );

G=PermutationGroup([(1,102,26,69,73),(2,103,27,70,74),(3,104,28,71,75),(4,105,29,72,76),(5,106,30,49,77),(6,107,31,50,78),(7,108,32,51,79),(8,109,33,52,80),(9,110,34,53,81),(10,111,35,54,82),(11,112,36,55,83),(12,113,37,56,84),(13,114,38,57,85),(14,115,39,58,86),(15,116,40,59,87),(16,117,41,60,88),(17,118,42,61,89),(18,119,43,62,90),(19,120,44,63,91),(20,97,45,64,92),(21,98,46,65,93),(22,99,47,66,94),(23,100,48,67,95),(24,101,25,68,96)], [(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),(25,39),(27,37),(28,48),(29,35),(30,46),(31,33),(32,44),(34,42),(36,40),(41,47),(43,45),(49,65),(50,52),(51,63),(53,61),(54,72),(55,59),(56,70),(58,68),(60,66),(62,64),(67,71),(74,84),(75,95),(76,82),(77,93),(78,80),(79,91),(81,89),(83,87),(86,96),(88,94),(90,92),(97,119),(98,106),(99,117),(100,104),(101,115),(103,113),(105,111),(107,109),(108,120),(110,118),(112,116)])

C5×C24⋊C2 is a maximal subgroup of
C24⋊D10  C4014D6  C408D6  Dic60⋊C2  C40.31D6  Dic6.D10  D30.4D4  C5×S3×SD16

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++++++++
imageC1C2C2C2C5C10C10C10S3D4D6SD16D12C5×S3C5×D4C24⋊C2S3×C10C5×SD16C5×D12C5×C24⋊C2
kernelC5×C24⋊C2C120C5×Dic6C5×D12C24⋊C2C24Dic6D12C40C30C20C15C10C8C6C5C4C3C2C1
# reps111144441112244448816

Matrix representation of C5×C24⋊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] >;

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

Subgroup lattice of C5×C24⋊C2 in TeX

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