direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C2×C12, C60⋊12C22, C30.39C23, C30⋊8(C2×C4), C20⋊3(C2×C6), (C2×C20)⋊5C6, (C2×C60)⋊12C2, C10⋊2(C2×C12), C15⋊9(C22×C4), C5⋊2(C22×C12), Dic5⋊3(C2×C6), (C2×Dic5)⋊5C6, D10.8(C2×C6), (C2×C6).37D10, C22.9(C6×D5), (C6×Dic5)⋊11C2, C10.2(C22×C6), (C22×D5).5C6, C6.39(C22×D5), (C2×C30).38C22, (C6×D5).27C22, (C3×Dic5)⋊10C22, C2.1(D5×C2×C6), (D5×C2×C6).8C2, (C2×C10).9(C2×C6), SmallGroup(240,156)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C12 |
Generators and relations for D5×C2×C12
G = < a,b,c,d | a2=b12=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 260 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C12, C3×Dic5, C60, C6×D5, C2×C30, C2×C4×D5, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, D10, C2×C12, C22×C6, C3×D5, C4×D5, C22×D5, C22×C12, C6×D5, C2×C4×D5, D5×C12, D5×C2×C6, D5×C2×C12
►On 120 points
Generators in S120
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 97)(35 98)(36 99)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)(73 118)(74 119)(75 120)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)
(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)
(1 104 46 119 88)(2 105 47 120 89)(3 106 48 109 90)(4 107 37 110 91)(5 108 38 111 92)(6 97 39 112 93)(7 98 40 113 94)(8 99 41 114 95)(9 100 42 115 96)(10 101 43 116 85)(11 102 44 117 86)(12 103 45 118 87)(13 35 62 80 56)(14 36 63 81 57)(15 25 64 82 58)(16 26 65 83 59)(17 27 66 84 60)(18 28 67 73 49)(19 29 68 74 50)(20 30 69 75 51)(21 31 70 76 52)(22 32 71 77 53)(23 33 72 78 54)(24 34 61 79 55)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 85)(11 86)(12 87)(13 56)(14 57)(15 58)(16 59)(17 60)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 82)(26 83)(27 84)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 109)(107 110)(108 111)
G:=sub<Sym(120)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,97)(35,98)(36,99)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,85)(60,86)(73,118)(74,119)(75,120)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117), (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), (1,104,46,119,88)(2,105,47,120,89)(3,106,48,109,90)(4,107,37,110,91)(5,108,38,111,92)(6,97,39,112,93)(7,98,40,113,94)(8,99,41,114,95)(9,100,42,115,96)(10,101,43,116,85)(11,102,44,117,86)(12,103,45,118,87)(13,35,62,80,56)(14,36,63,81,57)(15,25,64,82,58)(16,26,65,83,59)(17,27,66,84,60)(18,28,67,73,49)(19,29,68,74,50)(20,30,69,75,51)(21,31,70,76,52)(22,32,71,77,53)(23,33,72,78,54)(24,34,61,79,55), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,56)(14,57)(15,58)(16,59)(17,60)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,82)(26,83)(27,84)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,109)(107,110)(108,111)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,97)(35,98)(36,99)(37,71)(38,72)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,85)(60,86)(73,118)(74,119)(75,120)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117), (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), (1,104,46,119,88)(2,105,47,120,89)(3,106,48,109,90)(4,107,37,110,91)(5,108,38,111,92)(6,97,39,112,93)(7,98,40,113,94)(8,99,41,114,95)(9,100,42,115,96)(10,101,43,116,85)(11,102,44,117,86)(12,103,45,118,87)(13,35,62,80,56)(14,36,63,81,57)(15,25,64,82,58)(16,26,65,83,59)(17,27,66,84,60)(18,28,67,73,49)(19,29,68,74,50)(20,30,69,75,51)(21,31,70,76,52)(22,32,71,77,53)(23,33,72,78,54)(24,34,61,79,55), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,85)(11,86)(12,87)(13,56)(14,57)(15,58)(16,59)(17,60)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,82)(26,83)(27,84)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,109)(107,110)(108,111) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,97),(35,98),(36,99),(37,71),(38,72),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,85),(60,86),(73,118),(74,119),(75,120),(76,109),(77,110),(78,111),(79,112),(80,113),(81,114),(82,115),(83,116),(84,117)], [(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)], [(1,104,46,119,88),(2,105,47,120,89),(3,106,48,109,90),(4,107,37,110,91),(5,108,38,111,92),(6,97,39,112,93),(7,98,40,113,94),(8,99,41,114,95),(9,100,42,115,96),(10,101,43,116,85),(11,102,44,117,86),(12,103,45,118,87),(13,35,62,80,56),(14,36,63,81,57),(15,25,64,82,58),(16,26,65,83,59),(17,27,66,84,60),(18,28,67,73,49),(19,29,68,74,50),(20,30,69,75,51),(21,31,70,76,52),(22,32,71,77,53),(23,33,72,78,54),(24,34,61,79,55)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,85),(11,86),(12,87),(13,56),(14,57),(15,58),(16,59),(17,60),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,82),(26,83),(27,84),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(106,109),(107,110),(108,111)]])
D5×C2×C12 is a maximal subgroup of
C60.93D4 C30.7M4(2) D10.10D12 Dic3⋊C4⋊D5 D10⋊Dic6 (D5×C12)⋊C4 (C4×D5)⋊Dic3 C60.67D4 C60.68D4 D6⋊(C4×D5) C15⋊20(C4×D4) D6⋊C4⋊D5 D10⋊D12 C60⋊D4 C12⋊7D20 C60.59(C2×C4) (C2×C12)⋊6F5
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | ··· | 12H | 12I | ··· | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D5 | D10 | D10 | C3×D5 | C4×D5 | C6×D5 | C6×D5 | D5×C12 |
| kernel | D5×C2×C12 | D5×C12 | C6×Dic5 | C2×C60 | D5×C2×C6 | C2×C4×D5 | C6×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 16 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of D5×C2×C12 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 21 | 0 | 0 |
| 0 | 47 | 0 |
| 0 | 0 | 47 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 17 |
| 60 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,60],[21,0,0,0,47,0,0,0,47],[1,0,0,0,0,60,0,1,17],[60,0,0,0,0,1,0,1,0] >;
D5×C2×C12 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_{12} % in TeX
G:=Group("D5xC2xC12"); // GroupNames label
G:=SmallGroup(240,156);
// by ID
G=gap.SmallGroup(240,156);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,122,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations