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G = C5xD24order 240 = 24·3·5

Direct product of C5 and D24

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

Aliases: C5xD24, C15:6D8, C40:5S3, C120:6C2, C24:1C10, D12:1C10, C20.53D6, C30.29D4, C10.14D12, C60.65C22, C3:1(C5xD8), C8:1(C5xS3), C6.2(C5xD4), (C5xD12):7C2, C4.9(S3xC10), C2.4(C5xD12), C12.9(C2xC10), SmallGroup(240,52)

Series: Derived Chief Lower central Upper central

C1C12 — C5xD24
C1C3C6C12C60C5xD12 — C5xD24
C3C6C12 — C5xD24
C1C10C20C40

Generators and relations for C5xD24
 G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 136 in 44 conjugacy classes, 22 normal (18 characteristic)
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, D8, C2xC10, D12, C5xS3, C5xD4, D24, S3xC10, C5xD8, C5xD12, C5xD24
12C2
12C2
6C22
6C22
4S3
4S3
12C10
12C10
3D4
3D4
2D6
2D6
6C2xC10
6C2xC10
4C5xS3
4C5xS3
3D8
3C5xD4
3C5xD4
2S3xC10
2S3xC10
3C5xD8

Smallest permutation representation of C5xD24
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)]])

C5xD24 is a maximal subgroup of
C15:D16  C5:D48  C15:SD32  D24.D5  D24:D5  C40:5D6  D24:6D5  D24:7D5  D24:5D5  C5xS3xD8

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+++++++++
imageC1C2C2C5C10C10S3D4D6D8D12C5xS3C5xD4D24S3xC10C5xD8C5xD12C5xD24
kernelC5xD24C120C5xD12D24C24D12C40C30C20C15C10C8C6C5C4C3C2C1
# reps1124481112244448816

Matrix representation of C5xD24 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] >;

C5xD24 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 C5xD24 in TeX

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