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G = C5⋊D24order 240 = 24·3·5

The semidirect product of C5 and D24 acting via D24/D12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D24, C153D8, D121D5, D608C2, C30.3D4, C20.2D6, C10.6D12, C12.22D10, C60.15C22, C52C81S3, C31(D4⋊D5), C4.8(S3×D5), (C5×D12)⋊1C2, C6.1(C5⋊D4), C2.4(C5⋊D12), (C3×C52C8)⋊1C2, SmallGroup(240,15)

Series: Derived Chief Lower central Upper central

C1C60 — C5⋊D24
C1C5C15C30C60C3×C52C8 — C5⋊D24
C15C30C60 — C5⋊D24
C1C2C4

Generators and relations for C5⋊D24
 G = < a,b,c | a5=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >

12C2
60C2
6C22
30C22
4S3
20S3
12D5
12C10
3D4
5C8
15D4
2D6
10D6
6C2×C10
6D10
4D15
4C5×S3
15D8
5D12
5C24
3C5×D4
3D20
2D30
2S3×C10
5D24
3D4⋊D5

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

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

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

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

C5⋊D24 is a maximal subgroup of
D5×D24  C24⋊D10  D24⋊D5  C40.31D6  C20.60D12  D6036C22  C60.38D4  S3×D4⋊D5  D15⋊D8  D12.9D10  D125D10  D12⋊D10  D60⋊C22  Dic10.27D6  D12.D10
C5⋊D24 is a maximal quotient of
C5⋊D48  D24.D5  Dic12⋊D5  C5⋊Dic24  C10.D24  D6015C4  C60.8Q8

33 conjugacy classes

class 1 2A2B2C 3  4 5A5B 6 8A8B10A10B10C10D10E10F12A12B15A15B20A20B24A24B24C24D30A30B60A60B60C60D
order1222345568810101010101012121515202024242424303060606060
size111260222221010221212121222444410101010444444

33 irreducible representations

dim11112222222224444
type++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10D12C5⋊D4D24S3×D5D4⋊D5C5⋊D12C5⋊D24
kernelC5⋊D24C3×C52C8C5×D12D60C52C8C30D12C20C15C12C10C6C5C4C3C2C1
# reps11111121222442224

Matrix representation of C5⋊D24 in GL6(𝔽241)

100000
010000
00240100
005019000
000010
000001
,
01290000
1702190000
00692700
002017200
0000044
000011556
,
100000
1252400000
0051100
005119000
00002400
00001741

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,50,0,0,0,0,1,190,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,170,0,0,0,0,129,219,0,0,0,0,0,0,69,20,0,0,0,0,27,172,0,0,0,0,0,0,0,115,0,0,0,0,44,56],[1,125,0,0,0,0,0,240,0,0,0,0,0,0,51,51,0,0,0,0,1,190,0,0,0,0,0,0,240,174,0,0,0,0,0,1] >;

C5⋊D24 in GAP, Magma, Sage, TeX

C_5\rtimes D_{24}
% in TeX

G:=Group("C5:D24");
// GroupNames label

G:=SmallGroup(240,15);
// by ID

G=gap.SmallGroup(240,15);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,73,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊D24 in TeX

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