direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C22.F5, C5⋊C8⋊3D6, D6.F5⋊5C2, C5⋊4(S3×M4(2)), D6.9(C2×F5), C15⋊4(C2×M4(2)), C15⋊C8⋊3C22, C22.8(S3×F5), (C5×S3)⋊2M4(2), (S3×Dic5).5C4, (C22×S3).3F5, C6.23(C22×F5), C15⋊8M4(2)⋊3C2, C30.23(C22×C4), Dic5.28(C4×S3), (C2×Dic5).148D6, (C2×Dic15).10C4, Dic15.15(C2×C4), Dic5.35(C22×S3), (S3×Dic5).17C22, (C3×Dic5).33C23, (C6×Dic5).144C22, (S3×C5⋊C8)⋊5C2, C2.24(C2×S3×F5), (C3×C5⋊C8)⋊3C22, (S3×C2×C10).5C4, C10.23(S3×C2×C4), (C2×C6).6(C2×F5), C3⋊2(C2×C22.F5), (C2×C30).18(C2×C4), (C2×C10).19(C4×S3), (C2×S3×Dic5).12C2, (S3×C10).14(C2×C4), (C3×C22.F5)⋊2C2, (C3×Dic5).25(C2×C4), SmallGroup(480,1004)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 564 in 136 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2 [×4], C3, C4 [×4], C22, C22 [×4], C5, S3 [×2], S3, C6, C6, C8 [×4], C2×C4 [×6], C23, C10, C10 [×4], Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], Dic5 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C5×S3, C30, C30, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×2], C2×Dic5, C2×Dic5 [×5], C22×C10, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, C3×Dic5 [×2], Dic15 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C5⋊C8 [×2], C22.F5, C22.F5 [×3], C22×Dic5, S3×M4(2), C3×C5⋊C8 [×2], C15⋊C8 [×2], S3×Dic5 [×4], C6×Dic5, C2×Dic15, S3×C2×C10, C2×C22.F5, S3×C5⋊C8 [×2], D6.F5 [×2], C3×C22.F5, C15⋊8M4(2), C2×S3×Dic5, S3×C22.F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, F5, C4×S3 [×2], C22×S3, C2×M4(2), C2×F5 [×3], S3×C2×C4, C22.F5 [×2], C22×F5, S3×M4(2), S3×F5, C2×C22.F5, C2×S3×F5, S3×C22.F5
Generators and relations
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 >
►On 120 points
Generators in S120
(1 26 50)(2 27 51)(3 28 52)(4 29 53)(5 30 54)(6 31 55)(7 32 56)(8 25 49)(9 24 60)(10 17 61)(11 18 62)(12 19 63)(13 20 64)(14 21 57)(15 22 58)(16 23 59)(33 41 113)(34 42 114)(35 43 115)(36 44 116)(37 45 117)(38 46 118)(39 47 119)(40 48 120)(65 109 92)(66 110 93)(67 111 94)(68 112 95)(69 105 96)(70 106 89)(71 107 90)(72 108 91)(73 88 104)(74 81 97)(75 82 98)(76 83 99)(77 84 100)(78 85 101)(79 86 102)(80 87 103)
(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(65 109)(66 110)(67 111)(68 112)(69 105)(70 106)(71 107)(72 108)(73 104)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(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)(97 101)(99 103)(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 88 24)(2 81 89 17 114)(3 18 82 115 90)(4 116 19 91 83)(5 92 117 84 20)(6 85 93 21 118)(7 22 86 119 94)(8 120 23 95 87)(9 50 105 41 73)(10 42 51 74 106)(11 75 43 107 52)(12 108 76 53 44)(13 54 109 45 77)(14 46 55 78 110)(15 79 47 111 56)(16 112 80 49 48)(25 40 59 68 103)(26 69 33 104 60)(27 97 70 61 34)(28 62 98 35 71)(29 36 63 72 99)(30 65 37 100 64)(31 101 66 57 38)(32 58 102 39 67)
(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)| (1,26,50)(2,27,51)(3,28,52)(4,29,53)(5,30,54)(6,31,55)(7,32,56)(8,25,49)(9,24,60)(10,17,61)(11,18,62)(12,19,63)(13,20,64)(14,21,57)(15,22,58)(16,23,59)(33,41,113)(34,42,114)(35,43,115)(36,44,116)(37,45,117)(38,46,118)(39,47,119)(40,48,120)(65,109,92)(66,110,93)(67,111,94)(68,112,95)(69,105,96)(70,106,89)(71,107,90)(72,108,91)(73,88,104)(74,81,97)(75,82,98)(76,83,99)(77,84,100)(78,85,101)(79,86,102)(80,87,103), (9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(65,109)(66,110)(67,111)(68,112)(69,105)(70,106)(71,107)(72,108)(73,104)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(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)(97,101)(99,103)(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), (1,96,113,88,24)(2,81,89,17,114)(3,18,82,115,90)(4,116,19,91,83)(5,92,117,84,20)(6,85,93,21,118)(7,22,86,119,94)(8,120,23,95,87)(9,50,105,41,73)(10,42,51,74,106)(11,75,43,107,52)(12,108,76,53,44)(13,54,109,45,77)(14,46,55,78,110)(15,79,47,111,56)(16,112,80,49,48)(25,40,59,68,103)(26,69,33,104,60)(27,97,70,61,34)(28,62,98,35,71)(29,36,63,72,99)(30,65,37,100,64)(31,101,66,57,38)(32,58,102,39,67), (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( (1,26,50)(2,27,51)(3,28,52)(4,29,53)(5,30,54)(6,31,55)(7,32,56)(8,25,49)(9,24,60)(10,17,61)(11,18,62)(12,19,63)(13,20,64)(14,21,57)(15,22,58)(16,23,59)(33,41,113)(34,42,114)(35,43,115)(36,44,116)(37,45,117)(38,46,118)(39,47,119)(40,48,120)(65,109,92)(66,110,93)(67,111,94)(68,112,95)(69,105,96)(70,106,89)(71,107,90)(72,108,91)(73,88,104)(74,81,97)(75,82,98)(76,83,99)(77,84,100)(78,85,101)(79,86,102)(80,87,103), (9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(65,109)(66,110)(67,111)(68,112)(69,105)(70,106)(71,107)(72,108)(73,104)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(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)(97,101)(99,103)(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), (1,96,113,88,24)(2,81,89,17,114)(3,18,82,115,90)(4,116,19,91,83)(5,92,117,84,20)(6,85,93,21,118)(7,22,86,119,94)(8,120,23,95,87)(9,50,105,41,73)(10,42,51,74,106)(11,75,43,107,52)(12,108,76,53,44)(13,54,109,45,77)(14,46,55,78,110)(15,79,47,111,56)(16,112,80,49,48)(25,40,59,68,103)(26,69,33,104,60)(27,97,70,61,34)(28,62,98,35,71)(29,36,63,72,99)(30,65,37,100,64)(31,101,66,57,38)(32,58,102,39,67), (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([(1,26,50),(2,27,51),(3,28,52),(4,29,53),(5,30,54),(6,31,55),(7,32,56),(8,25,49),(9,24,60),(10,17,61),(11,18,62),(12,19,63),(13,20,64),(14,21,57),(15,22,58),(16,23,59),(33,41,113),(34,42,114),(35,43,115),(36,44,116),(37,45,117),(38,46,118),(39,47,119),(40,48,120),(65,109,92),(66,110,93),(67,111,94),(68,112,95),(69,105,96),(70,106,89),(71,107,90),(72,108,91),(73,88,104),(74,81,97),(75,82,98),(76,83,99),(77,84,100),(78,85,101),(79,86,102),(80,87,103)], [(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(65,109),(66,110),(67,111),(68,112),(69,105),(70,106),(71,107),(72,108),(73,104),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(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),(97,101),(99,103),(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)], [(1,96,113,88,24),(2,81,89,17,114),(3,18,82,115,90),(4,116,19,91,83),(5,92,117,84,20),(6,85,93,21,118),(7,22,86,119,94),(8,120,23,95,87),(9,50,105,41,73),(10,42,51,74,106),(11,75,43,107,52),(12,108,76,53,44),(13,54,109,45,77),(14,46,55,78,110),(15,79,47,111,56),(16,112,80,49,48),(25,40,59,68,103),(26,69,33,104,60),(27,97,70,61,34),(28,62,98,35,71),(29,36,63,72,99),(30,65,37,100,64),(31,101,66,57,38),(32,58,102,39,67)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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