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G = D2014D6order 480 = 25·3·5

8th semidirect product of D20 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2014D6, D609C22, Dic613D10, C60.54C23, C30.30C24, C1552+ 1+4, D30.15C23, Dic15.17C23, (C4×D5)⋊4D6, (D4×D5)⋊5S3, C5⋊D46D6, (S3×D20)⋊5C2, (D4×D15)⋊6C2, (C4×S3)⋊4D10, (C5×D4)⋊14D6, D411(S3×D5), C3⋊D46D10, (C3×D4)⋊14D10, D42S35D5, C53(D46D6), (C22×D5)⋊7D6, C15⋊Q813C22, D10⋊D64C2, C12.28D105C2, D20⋊S35C2, C33(D48D10), (S3×C20)⋊5C22, (C2×Dic3)⋊8D10, (C4×D15)⋊5C22, (D5×C12)⋊5C22, C157D46C22, (C2×C30).6C23, C6.30(C23×D5), Dic5.D65C2, D6.D103C2, (C3×D20)⋊10C22, (D4×C15)⋊12C22, C15⋊D415C22, C3⋊D2021C22, C5⋊D1215C22, C10.30(S3×C23), C20.54(C22×S3), D30.C22C22, (C5×Dic6)⋊9C22, (D5×Dic3)⋊3C22, D6.14(C22×D5), (C6×D5).46C23, C12.54(C22×D5), (S3×C10).15C23, D10.15(C22×S3), (C10×Dic3)⋊14C22, (C22×D15)⋊12C22, Dic3.27(C22×D5), (C3×Dic5).15C23, Dic5.16(C22×S3), (C5×Dic3).30C23, (C3×D4×D5)⋊7C2, C4.54(C2×S3×D5), (D5×C3⋊D4)⋊5C2, (D5×C2×C6)⋊9C22, (C2×S3×D5)⋊5C22, C22.6(C2×S3×D5), (C5×D42S3)⋊7C2, (C2×C3⋊D20)⋊20C2, C2.33(C22×S3×D5), (C3×C5⋊D4)⋊6C22, (C5×C3⋊D4)⋊6C22, (C2×C6).6(C22×D5), (C2×C10).6(C22×S3), SmallGroup(480,1102)

Series: Derived Chief Lower central Upper central

C1C30 — D2014D6
C1C5C15C30C6×D5C2×S3×D5D5×C3⋊D4 — D2014D6
C15C30 — D2014D6
C1C2D4

Generators and relations for D2014D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, cac-1=a11, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >

Subgroups: 1932 in 332 conjugacy classes, 108 normal (50 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×5], C22 [×2], C22 [×13], C5, S3 [×4], C6, C6 [×5], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], Dic3, Dic3 [×2], Dic3, C12, C12, D6, D6 [×7], C2×C6 [×2], C2×C6 [×5], C15, C2×D4 [×9], C4○D4 [×6], Dic5, Dic5, C20, C20 [×3], D10, D10 [×2], D10 [×9], C2×C10 [×2], C2×C10, Dic6, Dic6, C4×S3, C4×S3 [×3], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4, C3×D4 [×3], C22×S3 [×4], C22×C6 [×2], C5×S3, C3×D5 [×3], D15 [×3], C30, C30 [×2], 2+ 1+4, Dic10, C4×D5, C4×D5 [×5], D20, D20 [×8], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C22×D5 [×2], C22×D5 [×4], C4○D12 [×2], S3×D4 [×4], D42S3, D42S3 [×3], C2×C3⋊D4 [×4], C6×D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, C6×D5 [×2], C6×D5 [×2], S3×C10, D30, D30 [×2], D30 [×2], C2×C30 [×2], C2×D20 [×3], C4○D20 [×3], D4×D5, D4×D5 [×5], Q82D5 [×2], C5×C4○D4, D46D6, D5×Dic3 [×2], D30.C2 [×2], C15⋊D4, C3⋊D20, C3⋊D20 [×6], C5⋊D12, C15⋊Q8, D5×C12, C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], C4×D15, D60, C157D4 [×2], D4×C15, C2×S3×D5 [×2], D5×C2×C6 [×2], C22×D15 [×2], D48D10, D20⋊S3, D6.D10, C12.28D10, S3×D20, Dic5.D6 [×2], C2×C3⋊D20 [×2], D5×C3⋊D4 [×2], D10⋊D6 [×2], C3×D4×D5, C5×D42S3, D4×D15, D2014D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, D46D6, C2×S3×D5 [×3], D48D10, C22×S3×D5, D2014D6

Smallest permutation representation of D2014D6
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)
(1 38 82)(2 29 83 12 39 93)(3 40 84)(4 31 85 14 21 95)(5 22 86)(6 33 87 16 23 97)(7 24 88)(8 35 89 18 25 99)(9 26 90)(10 37 91 20 27 81)(11 28 92)(13 30 94)(15 32 96)(17 34 98)(19 36 100)(41 77 119 51 67 109)(42 68 120)(43 79 101 53 69 111)(44 70 102)(45 61 103 55 71 113)(46 72 104)(47 63 105 57 73 115)(48 74 106)(49 65 107 59 75 117)(50 76 108)(52 78 110)(54 80 112)(56 62 114)(58 64 116)(60 66 118)
(1 112)(2 101)(3 110)(4 119)(5 108)(6 117)(7 106)(8 115)(9 104)(10 113)(11 102)(12 111)(13 120)(14 109)(15 118)(16 107)(17 116)(18 105)(19 114)(20 103)(21 67)(22 76)(23 65)(24 74)(25 63)(26 72)(27 61)(28 70)(29 79)(30 68)(31 77)(32 66)(33 75)(34 64)(35 73)(36 62)(37 71)(38 80)(39 69)(40 78)(41 85)(42 94)(43 83)(44 92)(45 81)(46 90)(47 99)(48 88)(49 97)(50 86)(51 95)(52 84)(53 93)(54 82)(55 91)(56 100)(57 89)(58 98)(59 87)(60 96)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F10G10H12A12B15A15B20A20B20C20D20E20F20G20H20I20J30A30B30C30D30E30F60A60B
order122222222223444444556666666101010101010101012121515202020202020202020203030303030306060
size1122610101030303022666103022244101020202244441212420444466661212121244888888

57 irreducible representations

dim1111111111112222222222224444448
type++++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6D10D10D10D10D102+ 1+4S3×D5D46D6C2×S3×D5C2×S3×D5D48D10D2014D6
kernelD2014D6D20⋊S3D6.D10C12.28D10S3×D20Dic5.D6C2×C3⋊D20D5×C3⋊D4D10⋊D6C3×D4×D5C5×D42S3D4×D15D4×D5D42S3C4×D5D20C5⋊D4C5×D4C22×D5Dic6C4×S3C2×Dic3C3⋊D4C3×D4C15D4C5C4C22C3C1
# reps1111122221111211212224421222442

Matrix representation of D2014D6 in GL6(𝔽61)

6000000
0600000
000001
00006017
0006000
0014400
,
100000
010000
000001
000010
000100
001000
,
010000
6010000
001000
000100
0000600
0000060
,
27240000
51340000
00295400
00593200
00002954
00005932

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,60,44,0,0,0,60,0,0,0,0,1,17,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[0,60,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[27,51,0,0,0,0,24,34,0,0,0,0,0,0,29,59,0,0,0,0,54,32,0,0,0,0,0,0,29,59,0,0,0,0,54,32] >;

D2014D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{14}D_6
% in TeX

G:=Group("D20:14D6");
// GroupNames label

G:=SmallGroup(480,1102);
// by ID

G=gap.SmallGroup(480,1102);
# by ID

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

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,d*a*d=a^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations

׿
×
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