Copied to
clipboard

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

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

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

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

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

׿
×
𝔽