Copied to
clipboard

G = D20.9D6order 480 = 25·3·5

9th non-split extension by D20 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.9D6, D604C22, Dic610D10, C60.15C23, C3⋊C88D10, (D4×D5)⋊3S3, D4⋊D155C2, C3⋊D403C2, D4.S34D5, (C5×D4).7D6, (C4×D5).8D6, C36(D40⋊C2), (C6×D5).63D4, D4.11(S3×D5), C6.144(D4×D5), C12.28D101C2, C1517(C8⋊C22), C153C88C22, (C3×D4).22D10, C30.D43C2, C30.177(C2×D4), C53(D126C22), C20.32D64C2, C20.15(C22×S3), (C3×Dic5).14D4, (C5×Dic6)⋊4C22, (D5×C12).7C22, (C3×D20).5C22, (D4×C15).9C22, C12.15(C22×D5), D10.29(C3⋊D4), Dic5.23(C3⋊D4), (C3×D4×D5)⋊3C2, C4.15(C2×S3×D5), (C5×C3⋊C8)⋊8C22, (C5×D4.S3)⋊5C2, C2.26(D5×C3⋊D4), C10.47(C2×C3⋊D4), SmallGroup(480,567)

Series: Derived Chief Lower central Upper central

C1C60 — D20.9D6
C1C5C15C30C60D5×C12C12.28D10 — D20.9D6
C15C30C60 — D20.9D6
C1C2C4D4

Generators and relations for D20.9D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a8b, dbd=a3b, dcd=a10c-1 >

Subgroups: 812 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3, C6, C6 [×3], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C3×D5 [×2], D15, C30, C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], D4.S3, D4.S3, C4○D12, C6×D4, C5×Dic3, C3×Dic5, C60, C6×D5, C6×D5 [×3], D30, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D126C22, C5×C3⋊C8, C153C8, D30.C2, C3⋊D20, D5×C12, C3×D20, C3×C5⋊D4, C5×Dic6, D60, D4×C15, D5×C2×C6, D40⋊C2, C20.32D6, C3⋊D40, C30.D4, C5×D4.S3, D4⋊D15, C12.28D10, C3×D4×D5, D20.9D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D126C22, C2×S3×D5, D40⋊C2, D5×C3⋊D4, D20.9D6

Smallest permutation representation of D20.9D6
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)
(1 43 112)(2 52 113 10 44 101)(3 41 114 19 45 110)(4 50 115 8 46 119)(5 59 116 17 47 108)(6 48 117)(7 57 118 15 49 106)(9 55 120 13 51 104)(11 53 102)(12 42 103 20 54 111)(14 60 105 18 56 109)(16 58 107)(21 64 87 35 70 81)(22 73 88 24 71 90)(23 62 89 33 72 99)(25 80 91 31 74 97)(26 69 92 40 75 86)(27 78 93 29 76 95)(28 67 94 38 77 84)(30 65 96 36 79 82)(32 63 98 34 61 100)(37 68 83 39 66 85)
(1 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 84)(17 83)(18 82)(19 81)(20 100)(21 104)(22 103)(23 102)(24 101)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 64)(42 63)(43 62)(44 61)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D6E6F6G8A8B10A10B10C10D12A12B15A15B20A20B20C20D30A30B30C30D30E30F40A40B40C40D60A60B
order122222344455666666688101010101212151520202020303030303030404040406060
size114102060221012222441010202012602288420444424244488881212121288

45 irreducible representations

dim1111111122222222222244444448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C3⋊D4C3⋊D4C8⋊C22S3×D5D4×D5D126C22C2×S3×D5D40⋊C2D5×C3⋊D4D20.9D6
kernelD20.9D6C20.32D6C3⋊D40C30.D4C5×D4.S3D4⋊D15C12.28D10C3×D4×D5D4×D5C3×Dic5C6×D5D4.S3C4×D5D20C5×D4C3⋊C8Dic6C3×D4Dic5D10C15D4C6C5C4C3C2C1
# reps1111111111121112222212222442

Matrix representation of D20.9D6 in GL6(𝔽241)

100000
010000
001151188
000052240
0012412551240
002342331189
,
100000
010000
002405200
000100
00117118240240
0075901
,
2402400000
100000
001895200
002405200
0018905152
0024052191190
,
1122210000
1091290000
003333730
00135208108168
000021233
000010629

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,124,234,0,0,1,0,125,233,0,0,51,52,51,1,0,0,188,240,240,189],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,117,7,0,0,52,1,118,59,0,0,0,0,240,0,0,0,0,0,240,1],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,189,240,189,240,0,0,52,52,0,52,0,0,0,0,51,191,0,0,0,0,52,190],[112,109,0,0,0,0,221,129,0,0,0,0,0,0,33,135,0,0,0,0,33,208,0,0,0,0,73,108,212,106,0,0,0,168,33,29] >;

D20.9D6 in GAP, Magma, Sage, TeX

D_{20}._9D_6
% in TeX

G:=Group("D20.9D6");
// GroupNames label

G:=SmallGroup(480,567);
// by ID

G=gap.SmallGroup(480,567);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=a^10*c^-1>;
// generators/relations

׿
×
𝔽