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G = S3×D4.D5order 480 = 25·3·5

Direct product of S3 and D4.D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×D4.D5, Dic109D6, D12.7D10, C60.9C23, Dic303C22, (S3×D4).D5, C56(S3×SD16), C52C815D6, D4.1(S3×D5), D4.D153C2, (C5×S3)⋊2SD16, (C5×D4).18D6, (C3×D4).1D10, C1511(C2×SD16), C153C86C22, (S3×Dic10)⋊1C2, (S3×C10).32D4, (C4×S3).20D10, C20.D62C2, C30.171(C2×D4), D12.D52C2, C10.141(S3×D4), C20.9(C22×S3), C12.9(C22×D5), D6.20(C5⋊D4), (S3×C20).3C22, (C5×Dic3).12D4, (D4×C15).3C22, (C5×D12).3C22, Dic3.4(C5⋊D4), (C3×Dic10)⋊2C22, C4.9(C2×S3×D5), C32(C2×D4.D5), (C5×S3×D4).1C2, (S3×C52C8)⋊2C2, (C3×D4.D5)⋊1C2, C2.22(S3×C5⋊D4), C6.44(C2×C5⋊D4), (C3×C52C8)⋊5C22, SmallGroup(480,561)

Series: Derived Chief Lower central Upper central

C1C60 — S3×D4.D5
C1C5C15C30C60C3×Dic10S3×Dic10 — S3×D4.D5
C15C30C60 — S3×D4.D5
C1C2C4D4

Generators and relations for S3×D4.D5
 G = < a,b,c,d,e,f | a3=b2=c4=d2=e5=1, f2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 636 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], S3, C6, C6, C8 [×2], C2×C4 [×2], D4, D4 [×2], Q8 [×3], C23, C10, C10 [×4], Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20, C20, C2×C10 [×5], C3⋊C8, C24, Dic6 [×2], C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C5×S3, C30, C30, C2×SD16, C52C8, C52C8, Dic10, Dic10 [×2], C2×Dic5, C2×C20, C5×D4, C5×D4 [×2], C22×C10, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, S3×C10 [×3], C2×C30, C2×C52C8, D4.D5, D4.D5 [×3], C2×Dic10, D4×C10, S3×SD16, C3×C52C8, C153C8, S3×Dic5, C15⋊Q8, C3×Dic10, S3×C20, C5×D12, C5×C3⋊D4, Dic30, D4×C15, S3×C2×C10, C2×D4.D5, S3×C52C8, C20.D6, D12.D5, C3×D4.D5, D4.D15, S3×Dic10, C5×S3×D4, S3×D4.D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C22×S3, C2×SD16, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, D4.D5 [×2], C2×C5⋊D4, S3×SD16, C2×S3×D5, C2×D4.D5, S3×C5⋊D4, S3×D4.D5

Smallest permutation representation of S3×D4.D5
On 120 points
Generators in S120
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(66 71)(67 72)(68 73)(69 74)(70 75)(81 86)(82 87)(83 88)(84 89)(85 90)(96 101)(97 102)(98 103)(99 104)(100 105)(111 116)(112 117)(113 118)(114 119)(115 120)
(1 46 16 31)(2 47 17 32)(3 48 18 33)(4 49 19 34)(5 50 20 35)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(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 78 16 63)(2 77 17 62)(3 76 18 61)(4 80 19 65)(5 79 20 64)(6 83 21 68)(7 82 22 67)(8 81 23 66)(9 85 24 70)(10 84 25 69)(11 88 26 73)(12 87 27 72)(13 86 28 71)(14 90 29 75)(15 89 30 74)(31 108 46 93)(32 107 47 92)(33 106 48 91)(34 110 49 95)(35 109 50 94)(36 113 51 98)(37 112 52 97)(38 111 53 96)(39 115 54 100)(40 114 55 99)(41 118 56 103)(42 117 57 102)(43 116 58 101)(44 120 59 105)(45 119 60 104)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B8A8B8C8D10A10B10C10D10E10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D24A24B30A30B30C30D30E30F60A60B
order12222234444556688881010101010101010101010101010121215152020202024243030303030306060
size1133412226206022281010303022444466661212121244044441212202044888888

51 irreducible representations

dim1111111122222222222224444448
type++++++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6SD16D10D10D10C5⋊D4C5⋊D4S3×D4S3×D5D4.D5S3×SD16C2×S3×D5S3×C5⋊D4S3×D4.D5
kernelS3×D4.D5S3×C52C8C20.D6D12.D5C3×D4.D5D4.D15S3×Dic10C5×S3×D4D4.D5C5×Dic3S3×C10S3×D4C52C8Dic10C5×D4C5×S3C4×S3D12C3×D4Dic3D6C10D4S3C5C4C2C1
# reps1111111111121114222441242242

Matrix representation of S3×D4.D5 in GL6(𝔽241)

100000
010000
002399000
008100
000010
000001
,
100000
010000
001000
0023324000
000010
000001
,
240810000
23810000
001000
000100
00002400
00000240
,
24000000
23810000
001000
000100
000024024
000001
,
100000
010000
001000
000100
000087211
00000205
,
0930000
5700000
00240000
00024000
000011132
000021230

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,239,8,0,0,0,0,90,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,233,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,238,0,0,0,0,81,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,238,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,24,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,0,0,0,0,0,211,205],[0,57,0,0,0,0,93,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,11,21,0,0,0,0,132,230] >;

S3×D4.D5 in GAP, Magma, Sage, TeX

S_3\times D_4.D_5
% in TeX

G:=Group("S3xD4.D5");
// GroupNames label

G:=SmallGroup(480,561);
// by ID

G=gap.SmallGroup(480,561);
# by ID

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

G:=Group<a,b,c,d,e,f|a^3=b^2=c^4=d^2=e^5=1,f^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽