direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C24⋊C2, C40⋊13D6, C24⋊23D10, Dic6⋊7D10, C120⋊13C22, D10.22D12, D12.18D10, Dic5.5D12, C60.116C23, D60.30C22, Dic30⋊16C22, (C8×D5)⋊4S3, C8⋊11(S3×D5), C6.1(D4×D5), C3⋊1(D5×SD16), (D5×C24)⋊4C2, C5⋊2C8⋊23D6, C2.6(D5×D12), C30.1(C2×D4), C15⋊1(C2×SD16), (C3×D5)⋊1SD16, (D5×Dic6)⋊8C2, (C6×D5).40D4, (D5×D12).2C2, (C4×D5).76D6, C10.1(C2×D12), C24⋊D5⋊13C2, D12.D5⋊9C2, Dic6⋊D5⋊9C2, C20.64(C22×S3), (C3×Dic5).44D4, (C5×Dic6)⋊12C22, (C5×D12).20C22, (D5×C12).90C22, C12.139(C22×D5), C5⋊1(C2×C24⋊C2), C4.64(C2×S3×D5), (C5×C24⋊C2)⋊3C2, (C3×C5⋊2C8)⋊27C22, SmallGroup(480,323)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C24⋊C2
G = < a,b,c,d | a5=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >
Subgroups: 956 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×C24⋊C2, C3×C5⋊2C8, C120, D5×Dic3, C5⋊D12, C15⋊Q8, D5×C12, C5×Dic6, C5×D12, Dic30, D60, C2×S3×D5, D5×SD16, D12.D5, Dic6⋊D5, D5×C24, C5×C24⋊C2, C24⋊D5, D5×Dic6, D5×D12, D5×C24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, D12, C22×S3, C2×SD16, C22×D5, C24⋊C2, C2×D12, S3×D5, D4×D5, C2×C24⋊C2, C2×S3×D5, D5×SD16, D5×D12, D5×C24⋊C2
►On 120 points
Generators in S120
(1 59 82 120 43)(2 60 83 97 44)(3 61 84 98 45)(4 62 85 99 46)(5 63 86 100 47)(6 64 87 101 48)(7 65 88 102 25)(8 66 89 103 26)(9 67 90 104 27)(10 68 91 105 28)(11 69 92 106 29)(12 70 93 107 30)(13 71 94 108 31)(14 72 95 109 32)(15 49 96 110 33)(16 50 73 111 34)(17 51 74 112 35)(18 52 75 113 36)(19 53 76 114 37)(20 54 77 115 38)(21 55 78 116 39)(22 56 79 117 40)(23 57 80 118 41)(24 58 81 119 42)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 37)(26 48)(27 35)(28 46)(29 33)(30 44)(32 42)(34 40)(36 38)(39 47)(41 45)(49 69)(50 56)(51 67)(52 54)(53 65)(55 63)(57 61)(58 72)(60 70)(62 68)(64 66)(73 79)(74 90)(75 77)(76 88)(78 86)(80 84)(81 95)(83 93)(85 91)(87 89)(92 96)(97 107)(98 118)(99 105)(100 116)(101 103)(102 114)(104 112)(106 110)(109 119)(111 117)(113 115)
G:=sub<Sym(120)| (1,59,82,120,43)(2,60,83,97,44)(3,61,84,98,45)(4,62,85,99,46)(5,63,86,100,47)(6,64,87,101,48)(7,65,88,102,25)(8,66,89,103,26)(9,67,90,104,27)(10,68,91,105,28)(11,69,92,106,29)(12,70,93,107,30)(13,71,94,108,31)(14,72,95,109,32)(15,49,96,110,33)(16,50,73,111,34)(17,51,74,112,35)(18,52,75,113,36)(19,53,76,114,37)(20,54,77,115,38)(21,55,78,116,39)(22,56,79,117,40)(23,57,80,118,41)(24,58,81,119,42), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,37)(26,48)(27,35)(28,46)(29,33)(30,44)(32,42)(34,40)(36,38)(39,47)(41,45)(49,69)(50,56)(51,67)(52,54)(53,65)(55,63)(57,61)(58,72)(60,70)(62,68)(64,66)(73,79)(74,90)(75,77)(76,88)(78,86)(80,84)(81,95)(83,93)(85,91)(87,89)(92,96)(97,107)(98,118)(99,105)(100,116)(101,103)(102,114)(104,112)(106,110)(109,119)(111,117)(113,115)>;
G:=Group( (1,59,82,120,43)(2,60,83,97,44)(3,61,84,98,45)(4,62,85,99,46)(5,63,86,100,47)(6,64,87,101,48)(7,65,88,102,25)(8,66,89,103,26)(9,67,90,104,27)(10,68,91,105,28)(11,69,92,106,29)(12,70,93,107,30)(13,71,94,108,31)(14,72,95,109,32)(15,49,96,110,33)(16,50,73,111,34)(17,51,74,112,35)(18,52,75,113,36)(19,53,76,114,37)(20,54,77,115,38)(21,55,78,116,39)(22,56,79,117,40)(23,57,80,118,41)(24,58,81,119,42), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,97)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,37)(26,48)(27,35)(28,46)(29,33)(30,44)(32,42)(34,40)(36,38)(39,47)(41,45)(49,69)(50,56)(51,67)(52,54)(53,65)(55,63)(57,61)(58,72)(60,70)(62,68)(64,66)(73,79)(74,90)(75,77)(76,88)(78,86)(80,84)(81,95)(83,93)(85,91)(87,89)(92,96)(97,107)(98,118)(99,105)(100,116)(101,103)(102,114)(104,112)(106,110)(109,119)(111,117)(113,115) );
G=PermutationGroup([[(1,59,82,120,43),(2,60,83,97,44),(3,61,84,98,45),(4,62,85,99,46),(5,63,86,100,47),(6,64,87,101,48),(7,65,88,102,25),(8,66,89,103,26),(9,67,90,104,27),(10,68,91,105,28),(11,69,92,106,29),(12,70,93,107,30),(13,71,94,108,31),(14,72,95,109,32),(15,49,96,110,33),(16,50,73,111,34),(17,51,74,112,35),(18,52,75,113,36),(19,53,76,114,37),(20,54,77,115,38),(21,55,78,116,39),(22,56,79,117,40),(23,57,80,118,41),(24,58,81,119,42)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,97),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109)], [(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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,37),(26,48),(27,35),(28,46),(29,33),(30,44),(32,42),(34,40),(36,38),(39,47),(41,45),(49,69),(50,56),(51,67),(52,54),(53,65),(55,63),(57,61),(58,72),(60,70),(62,68),(64,66),(73,79),(74,90),(75,77),(76,88),(78,86),(80,84),(81,95),(83,93),(85,91),(87,89),(92,96),(97,107),(98,118),(99,105),(100,116),(101,103),(102,114),(104,112),(106,110),(109,119),(111,117),(113,115)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 12 | 60 | 2 | 2 | 10 | 12 | 60 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 10 | 10 | 2 | 2 | 24 | 24 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | SD16 | D10 | D10 | D10 | D12 | D12 | C24⋊C2 | S3×D5 | D4×D5 | C2×S3×D5 | D5×SD16 | D5×D12 | D5×C24⋊C2 |
| kernel | D5×C24⋊C2 | D12.D5 | Dic6⋊D5 | D5×C24 | C5×C24⋊C2 | C24⋊D5 | D5×Dic6 | D5×D12 | C8×D5 | C3×Dic5 | C6×D5 | C24⋊C2 | C5⋊2C8 | C40 | C4×D5 | C3×D5 | C24 | Dic6 | D12 | Dic5 | D10 | D5 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D5×C24⋊C2 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 189 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 203 | 73 | 0 | 0 | 0 | 0 |
| 208 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 123 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,189,240],[203,208,0,0,0,0,73,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,123,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D5×C24⋊C2 in GAP, Magma, Sage, TeX
D_5\times C_{24}\rtimes C_2 % in TeX
G:=Group("D5xC24:C2"); // GroupNames label
G:=SmallGroup(480,323);
// by ID
G=gap.SmallGroup(480,323);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,58,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^24=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations