Copied to
clipboard

## G = S3×C22.F5order 480 = 25·3·5

### Direct product of S3 and C22.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C22.F5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — S3×C5⋊C8 — S3×C22.F5
 Lower central C15 — C30 — S3×C22.F5
 Upper central C1 — C2 — C22

Generators and relations for S3×C22.F5
G = < a,b,c,d,e,f | a3=b2=c2=d2=e5=1, f4=d, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 564 in 136 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, S3, C6, C6, C8, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C5×S3, C30, C30, C2×M4(2), C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C3×Dic5, Dic15, S3×C10, S3×C10, C2×C30, C2×C5⋊C8, C22.F5, C22.F5, C22×Dic5, S3×M4(2), C3×C5⋊C8, C15⋊C8, S3×Dic5, C6×Dic5, C2×Dic15, S3×C2×C10, C2×C22.F5, S3×C5⋊C8, D6.F5, C3×C22.F5, C158M4(2), C2×S3×Dic5, S3×C22.F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, F5, C4×S3, C22×S3, C2×M4(2), C2×F5, S3×C2×C4, C22.F5, C22×F5, S3×M4(2), S3×F5, C2×C22.F5, C2×S3×F5, S3×C22.F5

Smallest permutation representation of S3×C22.F5
On 120 points
Generators in S120
(1 11 50)(2 12 51)(3 13 52)(4 14 53)(5 15 54)(6 16 55)(7 9 56)(8 10 49)(17 33 25)(18 34 26)(19 35 27)(20 36 28)(21 37 29)(22 38 30)(23 39 31)(24 40 32)(41 113 71)(42 114 72)(43 115 65)(44 116 66)(45 117 67)(46 118 68)(47 119 69)(48 120 70)(57 102 78)(58 103 79)(59 104 80)(60 97 73)(61 98 74)(62 99 75)(63 100 76)(64 101 77)(81 110 93)(82 111 94)(83 112 95)(84 105 96)(85 106 89)(86 107 90)(87 108 91)(88 109 92)
(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(57 78)(58 79)(59 80)(60 73)(61 74)(62 75)(63 76)(64 77)(81 110)(82 111)(83 112)(84 105)(85 106)(86 107)(87 108)(88 109)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(106 110)(108 112)(114 118)(116 120)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)
(1 96 113 20 99)(2 21 89 100 114)(3 101 22 115 90)(4 116 102 91 23)(5 92 117 24 103)(6 17 93 104 118)(7 97 18 119 94)(8 120 98 95 19)(9 73 34 69 82)(10 70 74 83 35)(11 84 71 36 75)(12 37 85 76 72)(13 77 38 65 86)(14 66 78 87 39)(15 88 67 40 79)(16 33 81 80 68)(25 110 59 46 55)(26 47 111 56 60)(27 49 48 61 112)(28 62 50 105 41)(29 106 63 42 51)(30 43 107 52 64)(31 53 44 57 108)(32 58 54 109 45)
(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)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 12A 12B 12C 15 24A 24B 24C 24D 30A 30B 30C order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 6 6 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 12 12 12 15 24 24 24 24 30 30 30 size 1 1 2 3 3 6 2 5 5 10 15 15 30 4 2 4 10 10 10 10 30 30 30 30 4 4 4 12 12 12 12 10 10 20 8 20 20 20 20 8 8 8

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 8 8 8 type + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 F5 C2×F5 C2×F5 C22.F5 S3×M4(2) S3×F5 C2×S3×F5 S3×C22.F5 kernel S3×C22.F5 S3×C5⋊C8 D6.F5 C3×C22.F5 C15⋊8M4(2) C2×S3×Dic5 S3×Dic5 C2×Dic15 S3×C2×C10 C22.F5 C5⋊C8 C2×Dic5 C5×S3 Dic5 C2×C10 C22×S3 D6 C2×C6 S3 C5 C22 C2 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 1 4 2 2 1 2 1 4 2 1 1 2

Matrix representation of S3×C22.F5 in GL8(𝔽241)

 240 240 65 0 0 0 0 0 1 0 0 176 0 0 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 176 0 0 0 0 1 0 0 176 0 0 0 0 0 0 1 240 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 37 37 0 0 0 0 0 1 37 37 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 189 1 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 0 240 1 0 0 0 0 3 0 188 52
,
 0 63 161 162 0 0 0 0 0 63 161 161 0 0 0 0 64 128 178 178 0 0 0 0 177 64 0 0 0 0 0 0 0 0 0 0 69 217 142 200 0 0 0 0 126 72 191 229 0 0 0 0 50 86 43 126 0 0 0 0 82 57 88 57

G:=sub<GL(8,GF(241))| [240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,65,0,0,1,0,0,0,0,0,176,240,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,176,176,240,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,37,37,240,0,0,0,0,0,37,37,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,189,240,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,188,0,0,0,0,0,0,1,52],[0,0,64,177,0,0,0,0,63,63,128,64,0,0,0,0,161,161,178,0,0,0,0,0,162,161,178,0,0,0,0,0,0,0,0,0,69,126,50,82,0,0,0,0,217,72,86,57,0,0,0,0,142,191,43,88,0,0,0,0,200,229,126,57] >;

S3×C22.F5 in GAP, Magma, Sage, TeX

S_3\times C_2^2.F_5
% in TeX

G:=Group("S3xC2^2.F5");
// GroupNames label

G:=SmallGroup(480,1004);
// by ID

G=gap.SmallGroup(480,1004);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,80,1356,9414,2379]);
// Polycyclic

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

׿
×
𝔽