metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊2D24, C15⋊3D8, D12⋊1D5, D60⋊8C2, C30.3D4, C20.2D6, C10.6D12, C12.22D10, C60.15C22, C5⋊2C8⋊1S3, C3⋊1(D4⋊D5), C4.8(S3×D5), (C5×D12)⋊1C2, C6.1(C5⋊D4), C2.4(C5⋊D12), (C3×C5⋊2C8)⋊1C2, SmallGroup(240,15)
Series: Derived ►Chief ►Lower central ►Upper central
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
6D10
4D15
15D8
5D12
5C24
3D20
2D30
5D24
Smallest permutation representation of C5⋊D24
►On 120 points
Generators in S120
(1 99 71 88 47)(2 48 89 72 100)(3 101 49 90 25)(4 26 91 50 102)(5 103 51 92 27)(6 28 93 52 104)(7 105 53 94 29)(8 30 95 54 106)(9 107 55 96 31)(10 32 73 56 108)(11 109 57 74 33)(12 34 75 58 110)(13 111 59 76 35)(14 36 77 60 112)(15 113 61 78 37)(16 38 79 62 114)(17 115 63 80 39)(18 40 81 64 116)(19 117 65 82 41)(20 42 83 66 118)(21 119 67 84 43)(22 44 85 68 120)(23 97 69 86 45)(24 46 87 70 98)
(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)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(41 113)(42 112)(43 111)(44 110)(45 109)(46 108)(47 107)(48 106)(49 94)(50 93)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 96)(72 95)
G:=sub<Sym(120)| (1,99,71,88,47)(2,48,89,72,100)(3,101,49,90,25)(4,26,91,50,102)(5,103,51,92,27)(6,28,93,52,104)(7,105,53,94,29)(8,30,95,54,106)(9,107,55,96,31)(10,32,73,56,108)(11,109,57,74,33)(12,34,75,58,110)(13,111,59,76,35)(14,36,77,60,112)(15,113,61,78,37)(16,38,79,62,114)(17,115,63,80,39)(18,40,81,64,116)(19,117,65,82,41)(20,42,83,66,118)(21,119,67,84,43)(22,44,85,68,120)(23,97,69,86,45)(24,46,87,70,98), (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)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(41,113)(42,112)(43,111)(44,110)(45,109)(46,108)(47,107)(48,106)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,96)(72,95)>;
G:=Group( (1,99,71,88,47)(2,48,89,72,100)(3,101,49,90,25)(4,26,91,50,102)(5,103,51,92,27)(6,28,93,52,104)(7,105,53,94,29)(8,30,95,54,106)(9,107,55,96,31)(10,32,73,56,108)(11,109,57,74,33)(12,34,75,58,110)(13,111,59,76,35)(14,36,77,60,112)(15,113,61,78,37)(16,38,79,62,114)(17,115,63,80,39)(18,40,81,64,116)(19,117,65,82,41)(20,42,83,66,118)(21,119,67,84,43)(22,44,85,68,120)(23,97,69,86,45)(24,46,87,70,98), (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)(25,105)(26,104)(27,103)(28,102)(29,101)(30,100)(31,99)(32,98)(33,97)(34,120)(35,119)(36,118)(37,117)(38,116)(39,115)(40,114)(41,113)(42,112)(43,111)(44,110)(45,109)(46,108)(47,107)(48,106)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,96)(72,95) );
G=PermutationGroup([[(1,99,71,88,47),(2,48,89,72,100),(3,101,49,90,25),(4,26,91,50,102),(5,103,51,92,27),(6,28,93,52,104),(7,105,53,94,29),(8,30,95,54,106),(9,107,55,96,31),(10,32,73,56,108),(11,109,57,74,33),(12,34,75,58,110),(13,111,59,76,35),(14,36,77,60,112),(15,113,61,78,37),(16,38,79,62,114),(17,115,63,80,39),(18,40,81,64,116),(19,117,65,82,41),(20,42,83,66,118),(21,119,67,84,43),(22,44,85,68,120),(23,97,69,86,45),(24,46,87,70,98)], [(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),(25,105),(26,104),(27,103),(28,102),(29,101),(30,100),(31,99),(32,98),(33,97),(34,120),(35,119),(36,118),(37,117),(38,116),(39,115),(40,114),(41,113),(42,112),(43,111),(44,110),(45,109),(46,108),(47,107),(48,106),(49,94),(50,93),(51,92),(52,91),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,96),(72,95)]])
C5⋊D24 is a maximal subgroup of
D5×D24 C24⋊D10 D24⋊D5 C40.31D6 C20.60D12 D60⋊36C22 C60.38D4 S3×D4⋊D5 D15⋊D8 D12.9D10 D12⋊5D10 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 D60⋊15C4 C60.8Q8
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | D12 | C5⋊D4 | D24 | S3×D5 | D4⋊D5 | C5⋊D12 | C5⋊D24 |
| kernel | C5⋊D24 | C3×C5⋊2C8 | C5×D12 | D60 | C5⋊2C8 | C30 | D12 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C5⋊D24 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 50 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 129 | 0 | 0 | 0 | 0 |
| 170 | 219 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 27 | 0 | 0 |
| 0 | 0 | 20 | 172 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 44 |
| 0 | 0 | 0 | 0 | 115 | 56 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 125 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 51 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 174 | 1 |
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