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G = S3×D4.D5order 480 = 25·3·5

Direct product of S3 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — S3×D4.D5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — S3×Dic10 — S3×D4.D5
 Lower central C15 — C30 — C60 — S3×D4.D5
 Upper central C1 — C2 — C4 — D4

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, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C5×S3, C30, C30, C2×SD16, C52C8, C52C8, Dic10, Dic10, C2×Dic5, C2×C20, C5×D4, C5×D4, 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, C2×C30, C2×C52C8, D4.D5, D4.D5, 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, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, C2×SD16, C5⋊D4, C22×D5, S3×D4, S3×D5, D4.D5, 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 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 15A 15B 20A 20B 20C 20D 24A 24B 30A 30B 30C 30D 30E 30F 60A 60B order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 24 24 30 30 30 30 30 30 60 60 size 1 1 3 3 4 12 2 2 6 20 60 2 2 2 8 10 10 30 30 2 2 4 4 4 4 6 6 6 6 12 12 12 12 4 40 4 4 4 4 12 12 20 20 4 4 8 8 8 8 8 8

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 C5⋊D4 C5⋊D4 S3×D4 S3×D5 D4.D5 S3×SD16 C2×S3×D5 S3×C5⋊D4 S3×D4.D5 kernel S3×D4.D5 S3×C5⋊2C8 C20.D6 D12.D5 C3×D4.D5 D4.D15 S3×Dic10 C5×S3×D4 D4.D5 C5×Dic3 S3×C10 S3×D4 C5⋊2C8 Dic10 C5×D4 C5×S3 C4×S3 D12 C3×D4 Dic3 D6 C10 D4 S3 C5 C4 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 1 2 4 2 2 4 2

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 239 90 0 0 0 0 8 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 0 0 0 0 0 233 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 240 81 0 0 0 0 238 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 0 0 0 0 0 238 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 24 0 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 1 0 0 0 0 0 0 87 211 0 0 0 0 0 205
,
 0 93 0 0 0 0 57 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 11 132 0 0 0 0 21 230

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

׿
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