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G = D5×C4○D12order 480 = 25·3·5

Direct product of D5 and C4○D12

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

Aliases: D5×C4○D12, D1227D10, Dic625D10, D6039C22, C30.18C24, D30.5C23, C60.138C23, Dic3036C22, Dic15.8C23, (C2×C20)⋊6D6, (C4×D5)⋊17D6, C15⋊Q89C22, (D5×D12)⋊13C2, (C4×S3)⋊12D10, (C2×C12)⋊27D10, C3⋊D412D10, (C2×C60)⋊6C22, (D5×Dic6)⋊13C2, (C2×Dic5)⋊22D6, C12.28D1013C2, D125D513C2, D6.5(C22×D5), C6.18(C23×D5), (S3×C20)⋊12C22, Dic3.D107C2, D6.D109C2, (C4×D15)⋊14C22, (C5×D12)⋊24C22, (D5×C12)⋊21C22, C3⋊D2011C22, C5⋊D1211C22, C15⋊D411C22, C157D413C22, (S3×C10).5C23, C10.18(S3×C23), D30.C27C22, D6011C211C2, (S3×Dic5)⋊7C22, (C22×D5).99D6, (C6×D5).43C23, C20.187(C22×S3), (C2×C30).237C23, (C5×Dic6)⋊22C22, (C6×Dic5)⋊27C22, D10.55(C22×S3), C12.187(C22×D5), Dic3.8(C22×D5), (C5×Dic3).8C23, (D5×Dic3).10C22, (C3×Dic5).43C23, Dic5.43(C22×S3), (C2×C4×D5)⋊6S3, (C4×S3×D5)⋊8C2, C31(D5×C4○D4), (D5×C2×C12)⋊1C2, C158(C2×C4○D4), C54(C2×C4○D12), (C2×C4)⋊12(S3×D5), (D5×C3⋊D4)⋊7C2, C4.160(C2×S3×D5), (C5×C4○D12)⋊5C2, C22.9(C2×S3×D5), (C3×D5)⋊1(C4○D4), (C2×S3×D5).5C22, C2.21(C22×S3×D5), (C5×C3⋊D4)⋊8C22, (D5×C2×C6).120C22, (C2×C10).10(C22×S3), (C2×C6).247(C22×D5), SmallGroup(480,1090)

Series: Derived Chief Lower central Upper central

C1C30 — D5×C4○D12
C1C5C15C30C6×D5C2×S3×D5C4×S3×D5 — D5×C4○D12
C15C30 — D5×C4○D12
C1C4C2×C4

Generators and relations for D5×C4○D12
 G = < a,b,c,d,e | a5=b2=c4=e2=1, d6=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >

Subgroups: 1628 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×4], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×2], D5 [×3], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6 [×3], C4×S3 [×2], C4×S3 [×6], D12, D12 [×3], C2×Dic3 [×2], C3⋊D4 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C5×S3 [×2], C3×D5 [×2], C3×D5, D15 [×2], C30, C30, C2×C4○D4, Dic10 [×3], C4×D5 [×4], C4×D5 [×6], D20 [×3], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5, C22×D5 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12, C4○D12 [×7], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×C4×D5, C2×C4×D5 [×2], C4○D20 [×3], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, C2×C4○D12, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], D5×C12 [×4], C6×Dic5, C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, C2×S3×D5 [×2], D5×C2×C6, D5×C4○D4, D5×Dic6, D6.D10 [×2], D125D5, C12.28D10, C4×S3×D5 [×2], D5×D12, Dic3.D10 [×2], D5×C3⋊D4 [×2], D5×C2×C12, C5×C4○D12, D6011C2, D5×C4○D12
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, C23×D5, C2×C4○D12, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D5×C4○D12

Smallest permutation representation of D5×C4○D12
On 120 points
Generators in S120
(1 105 59 110 25)(2 106 60 111 26)(3 107 49 112 27)(4 108 50 113 28)(5 97 51 114 29)(6 98 52 115 30)(7 99 53 116 31)(8 100 54 117 32)(9 101 55 118 33)(10 102 56 119 34)(11 103 57 120 35)(12 104 58 109 36)(13 47 64 75 94)(14 48 65 76 95)(15 37 66 77 96)(16 38 67 78 85)(17 39 68 79 86)(18 40 69 80 87)(19 41 70 81 88)(20 42 71 82 89)(21 43 72 83 90)(22 44 61 84 91)(23 45 62 73 92)(24 46 63 74 93)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(73 92)(74 93)(75 94)(76 95)(77 96)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 109)(105 110)(106 111)(107 112)(108 113)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 106 103 100)(98 107 104 101)(99 108 105 102)(109 118 115 112)(110 119 116 113)(111 120 117 114)
(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 87)(2 86)(3 85)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 98)(14 97)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 84)(34 83)(35 82)(36 81)(37 50)(38 49)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 120)(72 119)

G:=sub<Sym(120)| (1,105,59,110,25)(2,106,60,111,26)(3,107,49,112,27)(4,108,50,113,28)(5,97,51,114,29)(6,98,52,115,30)(7,99,53,116,31)(8,100,54,117,32)(9,101,55,118,33)(10,102,56,119,34)(11,103,57,120,35)(12,104,58,109,36)(13,47,64,75,94)(14,48,65,76,95)(15,37,66,77,96)(16,38,67,78,85)(17,39,68,79,86)(18,40,69,80,87)(19,41,70,81,88)(20,42,71,82,89)(21,43,72,83,90)(22,44,61,84,91)(23,45,62,73,92)(24,46,63,74,93), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,109)(105,110)(106,111)(107,112)(108,113), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,118,115,112)(110,119,116,113)(111,120,117,114), (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), (1,87)(2,86)(3,85)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,98)(14,97)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,84)(34,83)(35,82)(36,81)(37,50)(38,49)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,120)(72,119)>;

G:=Group( (1,105,59,110,25)(2,106,60,111,26)(3,107,49,112,27)(4,108,50,113,28)(5,97,51,114,29)(6,98,52,115,30)(7,99,53,116,31)(8,100,54,117,32)(9,101,55,118,33)(10,102,56,119,34)(11,103,57,120,35)(12,104,58,109,36)(13,47,64,75,94)(14,48,65,76,95)(15,37,66,77,96)(16,38,67,78,85)(17,39,68,79,86)(18,40,69,80,87)(19,41,70,81,88)(20,42,71,82,89)(21,43,72,83,90)(22,44,61,84,91)(23,45,62,73,92)(24,46,63,74,93), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,109)(105,110)(106,111)(107,112)(108,113), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,106,103,100)(98,107,104,101)(99,108,105,102)(109,118,115,112)(110,119,116,113)(111,120,117,114), (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), (1,87)(2,86)(3,85)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,98)(14,97)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,84)(34,83)(35,82)(36,81)(37,50)(38,49)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,120)(72,119) );

G=PermutationGroup([(1,105,59,110,25),(2,106,60,111,26),(3,107,49,112,27),(4,108,50,113,28),(5,97,51,114,29),(6,98,52,115,30),(7,99,53,116,31),(8,100,54,117,32),(9,101,55,118,33),(10,102,56,119,34),(11,103,57,120,35),(12,104,58,109,36),(13,47,64,75,94),(14,48,65,76,95),(15,37,66,77,96),(16,38,67,78,85),(17,39,68,79,86),(18,40,69,80,87),(19,41,70,81,88),(20,42,71,82,89),(21,43,72,83,90),(22,44,61,84,91),(23,45,62,73,92),(24,46,63,74,93)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,61),(23,62),(24,63),(73,92),(74,93),(75,94),(76,95),(77,96),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,109),(105,110),(106,111),(107,112),(108,113)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,106,103,100),(98,107,104,101),(99,108,105,102),(109,118,115,112),(110,119,116,113),(111,120,117,114)], [(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)], [(1,87),(2,86),(3,85),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,88),(13,98),(14,97),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,84),(34,83),(35,82),(36,81),(37,50),(38,49),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(61,118),(62,117),(63,116),(64,115),(65,114),(66,113),(67,112),(68,111),(69,110),(70,109),(71,120),(72,119)])

72 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F10G10H12A12B12C12D12E12F12G12H15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222222222344444444445566666661010101010101010121212121212121215152020202020202020202030···3060···60
size112556610303021125566103030222221010101022441212121222221010101044222244121212124···44···4

72 irreducible representations

dim111111111111222222222222244444
type++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6C4○D4D10D10D10D10D10C4○D12S3×D5C2×S3×D5C2×S3×D5D5×C4○D4D5×C4○D12
kernelD5×C4○D12D5×Dic6D6.D10D125D5C12.28D10C4×S3×D5D5×D12Dic3.D10D5×C3⋊D4D5×C2×C12C5×C4○D12D6011C2C2×C4×D5C4○D12C4×D5C2×Dic5C2×C20C22×D5C3×D5Dic6C4×S3D12C3⋊D4C2×C12D5C2×C4C4C22C3C1
# reps112112122111124111424242824248

Matrix representation of D5×C4○D12 in GL4(𝔽61) generated by

44100
166000
0010
0001
,
606000
0100
0010
0001
,
60000
06000
00110
00011
,
60000
06000
00210
006032
,
1000
0100
002530
002836
G:=sub<GL(4,GF(61))| [44,16,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,11,0,0,0,0,11],[60,0,0,0,0,60,0,0,0,0,21,60,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,25,28,0,0,30,36] >;

D5×C4○D12 in GAP, Magma, Sage, TeX

D_5\times C_4\circ D_{12}
% in TeX

G:=Group("D5xC4oD12");
// GroupNames label

G:=SmallGroup(480,1090);
// by ID

G=gap.SmallGroup(480,1090);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,346,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^2=1,d^6=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^5>;
// generators/relations

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×
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