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G = D12.D5order 240 = 24·3·5

1st non-split extension by D12 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.5D6, C30.8D4, C155SD16, D12.1D5, C10.7D12, Dic308C2, C12.23D10, C60.16C22, C52C82S3, C4.9(S3×D5), C53(C24⋊C2), C31(D4.D5), (C5×D12).1C2, C6.2(C5⋊D4), C2.5(C5⋊D12), (C3×C52C8)⋊2C2, SmallGroup(240,20)

Series: Derived Chief Lower central Upper central

C1C60 — D12.D5
C1C5C15C30C60C3×C52C8 — D12.D5
C15C30C60 — D12.D5
C1C2C4

Generators and relations for D12.D5
 G = < a,b,c,d | a12=b2=c5=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

12C2
6C22
30C4
4S3
12C10
3D4
5C8
15Q8
2D6
10Dic3
6Dic5
6C2×C10
4C5×S3
15SD16
5C24
5Dic6
3Dic10
3C5×D4
2S3×C10
2Dic15
5C24⋊C2
3D4.D5

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

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

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

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

D12.D5 is a maximal subgroup of
D5×C24⋊C2  D24⋊D5  Dic60⋊C2  D247D5  C20.60D12  D6036C22  D12.33D10  D30.8D4  S3×D4.D5  D2010D6  D30.11D4  D15⋊SD16  Dic10.26D6  D20.27D6  D30.44D4
D12.D5 is a maximal quotient of
C10.D24  Dic3015C4  C60.7Q8

33 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B10C10D10E10F12A12B15A15B20A20B24A24B24C24D30A30B60A60B60C60D
order1223445568810101010101012121515202024242424303060606060
size111222602221010221212121222444410101010444444

33 irreducible representations

dim11112222222224444
type+++++++++++-+-
imageC1C2C2C2S3D4D5D6SD16D10D12C5⋊D4C24⋊C2S3×D5D4.D5C5⋊D12D12.D5
kernelD12.D5C3×C52C8C5×D12Dic30C52C8C30D12C20C15C12C10C6C5C4C3C2C1
# reps11111121222442224

Matrix representation of D12.D5 in GL4(𝔽241) generated by

1000
0100
004399
00142142
,
1000
0100
009999
00198142
,
0100
2405100
0010
0001
,
17316000
666800
0066147
0094213
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,43,142,0,0,99,142],[1,0,0,0,0,1,0,0,0,0,99,198,0,0,99,142],[0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[173,66,0,0,160,68,0,0,0,0,66,94,0,0,147,213] >;

D12.D5 in GAP, Magma, Sage, TeX

D_{12}.D_5
% in TeX

G:=Group("D12.D5");
// GroupNames label

G:=SmallGroup(240,20);
// by ID

G=gap.SmallGroup(240,20);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,169,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^5=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D12.D5 in TeX

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