Copied to
clipboard

G = C5×D24order 240 = 24·3·5

Direct product of C5 and D24

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

Aliases: C5×D24, C156D8, C405S3, C1206C2, C241C10, D121C10, C20.53D6, C30.29D4, C10.14D12, C60.65C22, C31(C5×D8), C81(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

C1C12 — C5×D24
C1C3C6C12C60C5×D12 — C5×D24
C3C6C12 — C5×D24
C1C10C20C40

Generators and relations for C5×D24
 G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

12C2
12C2
6C22
6C22
4S3
4S3
12C10
12C10
3D4
3D4
2D6
2D6
6C2×C10
6C2×C10
4C5×S3
4C5×S3
3D8
3C5×D4
3C5×D4
2S3×C10
2S3×C10
3C5×D8

Smallest permutation representation of C5×D24
On 120 points
Generators in S120
(1 27 89 55 104)(2 28 90 56 105)(3 29 91 57 106)(4 30 92 58 107)(5 31 93 59 108)(6 32 94 60 109)(7 33 95 61 110)(8 34 96 62 111)(9 35 73 63 112)(10 36 74 64 113)(11 37 75 65 114)(12 38 76 66 115)(13 39 77 67 116)(14 40 78 68 117)(15 41 79 69 118)(16 42 80 70 119)(17 43 81 71 120)(18 44 82 72 97)(19 45 83 49 98)(20 46 84 50 99)(21 47 85 51 100)(22 48 86 52 101)(23 25 87 53 102)(24 26 88 54 103)
(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 28)(26 27)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(111 120)(112 119)(113 118)(114 117)(115 116)

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

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

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

C5×D24 is a maximal subgroup of
C15⋊D16  C5⋊D48  C15⋊SD32  D24.D5  D24⋊D5  C405D6  D246D5  D247D5  D245D5  C5×S3×D8

75 conjugacy classes

class 1 2A2B2C 3  4 5A5B5C5D 6 8A8B10A10B10C10D10E···10L12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12223455556881010101010···1012121515151520202020242424243030303040···4060···60120···120
size111212221111222111112···122222222222222222222···22···22···2

75 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C5C10C10S3D4D6D8D12C5×S3C5×D4D24S3×C10C5×D8C5×D12C5×D24
kernelC5×D24C120C5×D12D24C24D12C40C30C20C15C10C8C6C5C4C3C2C1
# reps1124481112244448816

Matrix representation of C5×D24 in GL2(𝔽241) generated by

910
091
,
127105
136232
,
232114
1059
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

Subgroup lattice of C5×D24 in TeX

׿
×
𝔽