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G = D12.9D10order 480 = 25·3·5

9th non-split extension by D12 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D12.9D10, D605C22, Dic1010D6, C60.20C23, (S3×D4)⋊3D5, C52C88D6, D4⋊D158C2, C5⋊D243C2, C56(Q83D6), D4.D54S3, D4.6(S3×D5), (C3×D4).8D10, (C4×S3).8D10, (C5×D4).23D6, D60⋊C22C2, C1519(C8⋊C22), (S3×C10).34D4, C10.146(S3×D4), C20.D64C2, C30.182(C2×D4), D6.Dic54C2, C153C810C22, D6.14(C5⋊D4), C33(D4.D10), (S3×C20).8C22, C20.20(C22×S3), (C5×Dic3).14D4, (C5×D12).6C22, C12.20(C22×D5), (C3×Dic10)⋊4C22, (D4×C15).14C22, Dic3.11(C5⋊D4), (C5×S3×D4)⋊3C2, C4.20(C2×S3×D5), (C3×D4.D5)⋊6C2, C2.27(S3×C5⋊D4), C6.49(C2×C5⋊D4), (C3×C52C8)⋊8C22, SmallGroup(480,572)

Series: Derived Chief Lower central Upper central

C1C60 — D12.9D10
C1C5C15C30C60C3×Dic10D60⋊C2 — D12.9D10
C15C30C60 — D12.9D10
C1C2C4D4

Generators and relations for D12.9D10
 G = < a,b,c,d | a12=b2=d2=1, c10=a6, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c9 >

Subgroups: 732 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5, C10, C10 [×3], Dic3, C12, C12, D6, D6 [×4], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10 [×5], C3⋊C8, C24, C4×S3, C4×S3, D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], D15, C30, C30, C8⋊C22, C52C8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4 [×2], C22×C10, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10 [×3], D30, C2×C30, C4.Dic5, D4⋊D5 [×2], D4.D5, D4.D5, C4○D20, D4×C10, Q83D6, C3×C52C8, C153C8, D30.C2, C5⋊D12, C3×Dic10, S3×C20, C5×D12, C5×C3⋊D4, D60, D4×C15, S3×C2×C10, D4.D10, D6.Dic5, C5⋊D24, C20.D6, C3×D4.D5, D4⋊D15, D60⋊C2, C5×S3×D4, D12.9D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q83D6, C2×S3×D5, D4.D10, S3×C5⋊D4, D12.9D10

Smallest permutation representation of D12.9D10
On 120 points
Generators in S120
(1 53 48 6 92 87 11 43 58 16 82 97)(2 88 83 7 59 54 12 98 93 17 49 44)(3 55 50 8 94 89 13 45 60 18 84 99)(4 90 85 9 41 56 14 100 95 19 51 46)(5 57 52 10 96 91 15 47 42 20 86 81)(21 66 71 36 106 111 31 76 61 26 116 101)(22 112 117 37 62 67 32 102 107 27 72 77)(23 68 73 38 108 113 33 78 63 28 118 103)(24 114 119 39 64 69 34 104 109 29 74 79)(25 70 75 40 110 115 35 80 65 30 120 105)
(1 97)(2 54)(3 99)(4 56)(5 81)(6 58)(7 83)(8 60)(9 85)(10 42)(11 87)(12 44)(13 89)(14 46)(15 91)(16 48)(17 93)(18 50)(19 95)(20 52)(21 71)(22 107)(23 73)(24 109)(25 75)(26 111)(27 77)(28 113)(29 79)(30 115)(31 61)(32 117)(33 63)(34 119)(35 65)(36 101)(37 67)(38 103)(39 69)(40 105)(41 90)(43 92)(45 94)(47 96)(49 98)(51 100)(53 82)(55 84)(57 86)(59 88)(102 112)(104 114)(106 116)(108 118)(110 120)
(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 38)(2 27)(3 36)(4 25)(5 34)(6 23)(7 32)(8 21)(9 30)(10 39)(11 28)(12 37)(13 26)(14 35)(15 24)(16 33)(17 22)(18 31)(19 40)(20 29)(41 65)(42 74)(43 63)(44 72)(45 61)(46 70)(47 79)(48 68)(49 77)(50 66)(51 75)(52 64)(53 73)(54 62)(55 71)(56 80)(57 69)(58 78)(59 67)(60 76)(81 104)(82 113)(83 102)(84 111)(85 120)(86 109)(87 118)(88 107)(89 116)(90 105)(91 114)(92 103)(93 112)(94 101)(95 110)(96 119)(97 108)(98 117)(99 106)(100 115)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D10E10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D24A24B30A30B30C30D30E30F60A60B
order12222234445566881010101010101010101010101010121215152020202024243030303030306060
size11461260226202228206022444466661212121244044441212202044888888

48 irreducible representations

dim1111111122222222222244444448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C5⋊D4C5⋊D4C8⋊C22S3×D4S3×D5Q83D6C2×S3×D5D4.D10S3×C5⋊D4D12.9D10
kernelD12.9D10D6.Dic5C5⋊D24C20.D6C3×D4.D5D4⋊D15D60⋊C2C5×S3×D4D4.D5C5×Dic3S3×C10S3×D4C52C8Dic10C5×D4C4×S3D12C3×D4Dic3D6C15C10D4C5C4C3C2C1
# reps1111111111121112224411222442

Matrix representation of D12.9D10 in GL6(𝔽241)

100000
010000
001122
0024002390
00240240240240
001010
,
100000
010000
001122
0002400239
0000240240
000001
,
51510000
19010000
001020
00240240239239
0024002400
001111
,
231590000
31100000
00918918
0092329232
00116232232223
001161252329

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,240,240,1,0,0,1,0,240,0,0,0,2,239,240,1,0,0,2,0,240,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,240,0,0,0,0,2,0,240,0,0,0,2,239,240,1],[51,190,0,0,0,0,51,1,0,0,0,0,0,0,1,240,240,1,0,0,0,240,0,1,0,0,2,239,240,1,0,0,0,239,0,1],[231,31,0,0,0,0,59,10,0,0,0,0,0,0,9,9,116,116,0,0,18,232,232,125,0,0,9,9,232,232,0,0,18,232,223,9] >;

D12.9D10 in GAP, Magma, Sage, TeX

D_{12}._9D_{10}
% in TeX

G:=Group("D12.9D10");
// GroupNames label

G:=SmallGroup(480,572);
// by ID

G=gap.SmallGroup(480,572);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,675,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^10=a^6,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^7*b,d*c*d=c^9>;
// generators/relations

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