Copied to
clipboard

G = D5×C2×C12order 240 = 24·3·5

Direct product of C2×C12 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×C2×C12, C6012C22, C30.39C23, C308(C2×C4), C203(C2×C6), (C2×C20)⋊5C6, (C2×C60)⋊12C2, C102(C2×C12), C159(C22×C4), C52(C22×C12), Dic53(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

C1C5 — D5×C2×C12
C1C5C10C30C6×D5D5×C2×C6 — D5×C2×C12
C5 — D5×C2×C12
C1C2×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

Smallest permutation representation of 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)  C1520(C4×D4)  D6⋊C4⋊D5  D10⋊D12  C60⋊D4  C127D20  C60.59(C2×C4)  (C2×C12)⋊6F5

96 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H5A5B6A···6F6G···6N10A···10F12A···12H12I···12P15A15B15C15D20A···20H30A···30L60A···60P
order122222223344444444556···66···610···1012···1212···121515151520···2030···3060···60
size111155551111115555221···15···52···21···15···522222···22···22···2

96 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12D5D10D10C3×D5C4×D5C6×D5C6×D5D5×C12
kernelD5×C2×C12D5×C12C6×Dic5C2×C60D5×C2×C6C2×C4×D5C6×D5C4×D5C2×Dic5C2×C20C22×D5D10C2×C12C12C2×C6C2×C4C6C4C22C2
# reps1411128822216242488416

Matrix representation of D5×C2×C12 in GL3(𝔽61) generated by

6000
0600
0060
,
2100
0470
0047
,
100
001
06017
,
6000
001
010
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

׿
×
𝔽