metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊2S3, C24⋊2D5, C8⋊2D15, C120⋊2C2, C2.3D60, C4.8D30, C6.1D20, C15⋊7SD16, D60.1C2, C20.43D6, C30.19D4, C10.1D12, Dic30⋊1C2, C12.43D10, C60.50C22, C5⋊1(C24⋊C2), C3⋊1(C40⋊C2), SmallGroup(240,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊D5
G = < a,b,c | a8=b15=c2=1, ab=ba, cac=a3, cbc=b-1 >
Smallest permutation representation of C24⋊D5
►On 120 points
Generators in S120
(1 110 47 84 17 92 39 61)(2 111 48 85 18 93 40 62)(3 112 49 86 19 94 41 63)(4 113 50 87 20 95 42 64)(5 114 51 88 21 96 43 65)(6 115 52 89 22 97 44 66)(7 116 53 90 23 98 45 67)(8 117 54 76 24 99 31 68)(9 118 55 77 25 100 32 69)(10 119 56 78 26 101 33 70)(11 120 57 79 27 102 34 71)(12 106 58 80 28 103 35 72)(13 107 59 81 29 104 36 73)(14 108 60 82 30 105 37 74)(15 109 46 83 16 91 38 75)
(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 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(61 91)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 120)(89 119)(90 118)
G:=sub<Sym(120)| (1,110,47,84,17,92,39,61)(2,111,48,85,18,93,40,62)(3,112,49,86,19,94,41,63)(4,113,50,87,20,95,42,64)(5,114,51,88,21,96,43,65)(6,115,52,89,22,97,44,66)(7,116,53,90,23,98,45,67)(8,117,54,76,24,99,31,68)(9,118,55,77,25,100,32,69)(10,119,56,78,26,101,33,70)(11,120,57,79,27,102,34,71)(12,106,58,80,28,103,35,72)(13,107,59,81,29,104,36,73)(14,108,60,82,30,105,37,74)(15,109,46,83,16,91,38,75), (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)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(61,91)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,120)(89,119)(90,118)>;
G:=Group( (1,110,47,84,17,92,39,61)(2,111,48,85,18,93,40,62)(3,112,49,86,19,94,41,63)(4,113,50,87,20,95,42,64)(5,114,51,88,21,96,43,65)(6,115,52,89,22,97,44,66)(7,116,53,90,23,98,45,67)(8,117,54,76,24,99,31,68)(9,118,55,77,25,100,32,69)(10,119,56,78,26,101,33,70)(11,120,57,79,27,102,34,71)(12,106,58,80,28,103,35,72)(13,107,59,81,29,104,36,73)(14,108,60,82,30,105,37,74)(15,109,46,83,16,91,38,75), (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)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(61,91)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,120)(89,119)(90,118) );
G=PermutationGroup([[(1,110,47,84,17,92,39,61),(2,111,48,85,18,93,40,62),(3,112,49,86,19,94,41,63),(4,113,50,87,20,95,42,64),(5,114,51,88,21,96,43,65),(6,115,52,89,22,97,44,66),(7,116,53,90,23,98,45,67),(8,117,54,76,24,99,31,68),(9,118,55,77,25,100,32,69),(10,119,56,78,26,101,33,70),(11,120,57,79,27,102,34,71),(12,106,58,80,28,103,35,72),(13,107,59,81,29,104,36,73),(14,108,60,82,30,105,37,74),(15,109,46,83,16,91,38,75)], [(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),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(61,91),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109),(85,108),(86,107),(87,106),(88,120),(89,119),(90,118)]])
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
Dic30⋊8C4 C120⋊10C4 D60⋊8C4
63 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 60 | 2 | 2 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | D12 | D15 | D20 | C24⋊C2 | D30 | C40⋊C2 | D60 | C24⋊D5 |
| kernel | C24⋊D5 | C120 | Dic30 | D60 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C24⋊D5 ►in GL2(𝔽241) generated by
| 132 | 57 |
| 184 | 147 |
| 94 | 110 |
| 131 | 161 |
| 1 | 190 |
| 0 | 240 |
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
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