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G = C24⋊D5order 240 = 24·3·5

2nd semidirect product of C24 and D5 acting via D5/C5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C402S3, C242D5, C82D15, C1202C2, C2.3D60, C4.8D30, C6.1D20, C157SD16, D60.1C2, C20.43D6, C30.19D4, C10.1D12, Dic301C2, C12.43D10, C60.50C22, C51(C24⋊C2), C31(C40⋊C2), SmallGroup(240,67)

Series: Derived Chief Lower central Upper central

C1C60 — C24⋊D5
C1C5C15C30C60D60 — C24⋊D5
C15C30C60 — C24⋊D5
C1C2C4C8

Generators and relations for C24⋊D5
 G = < a,b,c | a8=b15=c2=1, ab=ba, cac=a3, cbc=b-1 >

60C2
30C22
30C4
20S3
12D5
15Q8
15D4
10Dic3
10D6
6D10
6Dic5
4D15
15SD16
5Dic6
5D12
3D20
3Dic10
2D30
2Dic15
5C24⋊C2
3C40⋊C2

Smallest permutation representation of C24⋊D5
On 120 points
Generators in S120
(1 114 47 88 17 92 32 73)(2 115 48 89 18 93 33 74)(3 116 49 90 19 94 34 75)(4 117 50 76 20 95 35 61)(5 118 51 77 21 96 36 62)(6 119 52 78 22 97 37 63)(7 120 53 79 23 98 38 64)(8 106 54 80 24 99 39 65)(9 107 55 81 25 100 40 66)(10 108 56 82 26 101 41 67)(11 109 57 83 27 102 42 68)(12 110 58 84 28 103 43 69)(13 111 59 85 29 104 44 70)(14 112 60 86 30 105 45 71)(15 113 46 87 16 91 31 72)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 47)(32 46)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 105)(75 104)(76 110)(77 109)(78 108)(79 107)(80 106)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)

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

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

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

C24⋊D5 is a maximal subgroup of
D5×C24⋊C2  D24⋊D5  S3×C40⋊C2  D40⋊S3  C24.2D10  Dic20⋊S3  C40.31D6  D6.1D20  C40.69D6  C8⋊D30  C8.D30  D8⋊D15  SD16×D15  D4.5D30  Q16⋊D15
C24⋊D5 is a maximal quotient of
Dic308C4  C12010C4  D608C4

63 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12234455688101012121515151520202020242424243030303040···4060···60120···120
size1160226022222222222222222222222222···22···22···2

63 irreducible representations

dim111122222222222222
type++++++++++++++
imageC1C2C2C2S3D4D5D6SD16D10D12D15D20C24⋊C2D30C40⋊C2D60C24⋊D5
kernelC24⋊D5C120Dic30D60C40C30C24C20C15C12C10C8C6C5C4C3C2C1
# reps1111112122244448816

Matrix representation of C24⋊D5 in GL2(𝔽241) generated by

13257
184147
,
94110
131161
,
1190
0240
G:=sub<GL(2,GF(241))| [132,184,57,147],[94,131,110,161],[1,0,190,240] >;

C24⋊D5 in GAP, Magma, Sage, TeX

C_{24}\rtimes D_5
% in TeX

G:=Group("C24:D5");
// GroupNames label

G:=SmallGroup(240,67);
// by ID

G=gap.SmallGroup(240,67);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,31,218,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^8=b^15=c^2=1,a*b=b*a,c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C24⋊D5 in TeX

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