direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D5⋊M4(2), D5⋊C8⋊4C6, C4.F5⋊5C6, D5⋊(C3×M4(2)), C5⋊2(C6×M4(2)), C4.20(C6×F5), (C2×C20).8C12, (C2×C60).19C4, C60.73(C2×C4), (C4×D5).8C12, (C2×C12).19F5, C12.73(C2×F5), C22.F5⋊3C6, C20.20(C2×C12), C22.6(C6×F5), (D5×C12).15C4, (C3×D5)⋊5M4(2), C15⋊14(C2×M4(2)), C6.47(C22×F5), D10.14(C2×C12), C10.3(C22×C12), C30.85(C22×C4), (C22×D5).9C12, Dic5.16(C2×C12), (D5×C12).132C22, Dic5.11(C22×C6), (C3×Dic5).71C23, (C6×Dic5).278C22, C5⋊C8⋊2(C2×C6), C2.5(C2×C6×F5), (C3×D5⋊C8)⋊9C2, (D5×C2×C6).19C4, (C2×C4×D5).15C6, (C2×C4).8(C3×F5), (C3×C5⋊C8)⋊12C22, (D5×C2×C12).36C2, (C3×C4.F5)⋊11C2, (C2×C6).30(C2×F5), (C2×C30).58(C2×C4), (C4×D5).30(C2×C6), (C6×D5).63(C2×C4), (C2×C10).15(C2×C12), (C3×C22.F5)⋊7C2, (C2×Dic5).55(C2×C6), (C3×Dic5).72(C2×C4), SmallGroup(480,1049)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 392 in 136 conjugacy classes, 72 normal (48 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C24 [×4], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C2×M4(2), C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C24 [×2], C3×M4(2) [×4], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×2], C2×C4×D5, C6×M4(2), C3×C5⋊C8 [×4], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, D5⋊M4(2), C3×D5⋊C8 [×2], C3×C4.F5 [×2], C3×C22.F5 [×2], D5×C2×C12, C3×D5⋊M4(2)
Quotients:
C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], M4(2) [×2], C22×C4, F5, C2×C12 [×6], C22×C6, C2×M4(2), C2×F5 [×3], C3×M4(2) [×2], C22×C12, C3×F5, C22×F5, C6×M4(2), C6×F5 [×3], D5⋊M4(2), C2×C6×F5, C3×D5⋊M4(2)
Generators and relations
G = < a,b,c,d,e | a3=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=b3, be=eb, dcd-1=b2c, ce=ec, ede=d5 >
►On 120 points
Generators in S120
(1 104 70)(2 97 71)(3 98 72)(4 99 65)(5 100 66)(6 101 67)(7 102 68)(8 103 69)(9 82 111)(10 83 112)(11 84 105)(12 85 106)(13 86 107)(14 87 108)(15 88 109)(16 81 110)(17 95 116)(18 96 117)(19 89 118)(20 90 119)(21 91 120)(22 92 113)(23 93 114)(24 94 115)(25 54 75)(26 55 76)(27 56 77)(28 49 78)(29 50 79)(30 51 80)(31 52 73)(32 53 74)(33 47 63)(34 48 64)(35 41 57)(36 42 58)(37 43 59)(38 44 60)(39 45 61)(40 46 62)
(1 108 22 42 54)(2 43 109 55 23)(3 56 44 24 110)(4 17 49 111 45)(5 112 18 46 50)(6 47 105 51 19)(7 52 48 20 106)(8 21 53 107 41)(9 61 99 95 78)(10 96 62 79 100)(11 80 89 101 63)(12 102 73 64 90)(13 57 103 91 74)(14 92 58 75 104)(15 76 93 97 59)(16 98 77 60 94)(25 70 87 113 36)(26 114 71 37 88)(27 38 115 81 72)(28 82 39 65 116)(29 66 83 117 40)(30 118 67 33 84)(31 34 119 85 68)(32 86 35 69 120)
(1 54)(2 23)(3 110)(4 45)(5 50)(6 19)(7 106)(8 41)(9 95)(10 62)(12 102)(13 91)(14 58)(16 98)(17 111)(20 52)(21 107)(24 56)(25 70)(26 37)(27 115)(29 66)(30 33)(31 119)(35 69)(36 87)(39 65)(40 83)(42 108)(43 55)(46 112)(47 51)(57 103)(59 76)(61 99)(63 80)(67 118)(68 85)(71 114)(72 81)(73 90)(75 104)(77 94)(79 100)(82 116)(86 120)(89 101)(93 97)
(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 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(73 77)(75 79)(81 85)(83 87)(90 94)(92 96)(98 102)(100 104)(106 110)(108 112)(113 117)(115 119)
G:=sub<Sym(120)| (1,104,70)(2,97,71)(3,98,72)(4,99,65)(5,100,66)(6,101,67)(7,102,68)(8,103,69)(9,82,111)(10,83,112)(11,84,105)(12,85,106)(13,86,107)(14,87,108)(15,88,109)(16,81,110)(17,95,116)(18,96,117)(19,89,118)(20,90,119)(21,91,120)(22,92,113)(23,93,114)(24,94,115)(25,54,75)(26,55,76)(27,56,77)(28,49,78)(29,50,79)(30,51,80)(31,52,73)(32,53,74)(33,47,63)(34,48,64)(35,41,57)(36,42,58)(37,43,59)(38,44,60)(39,45,61)(40,46,62), (1,108,22,42,54)(2,43,109,55,23)(3,56,44,24,110)(4,17,49,111,45)(5,112,18,46,50)(6,47,105,51,19)(7,52,48,20,106)(8,21,53,107,41)(9,61,99,95,78)(10,96,62,79,100)(11,80,89,101,63)(12,102,73,64,90)(13,57,103,91,74)(14,92,58,75,104)(15,76,93,97,59)(16,98,77,60,94)(25,70,87,113,36)(26,114,71,37,88)(27,38,115,81,72)(28,82,39,65,116)(29,66,83,117,40)(30,118,67,33,84)(31,34,119,85,68)(32,86,35,69,120), (1,54)(2,23)(3,110)(4,45)(5,50)(6,19)(7,106)(8,41)(9,95)(10,62)(12,102)(13,91)(14,58)(16,98)(17,111)(20,52)(21,107)(24,56)(25,70)(26,37)(27,115)(29,66)(30,33)(31,119)(35,69)(36,87)(39,65)(40,83)(42,108)(43,55)(46,112)(47,51)(57,103)(59,76)(61,99)(63,80)(67,118)(68,85)(71,114)(72,81)(73,90)(75,104)(77,94)(79,100)(82,116)(86,120)(89,101)(93,97), (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), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(73,77)(75,79)(81,85)(83,87)(90,94)(92,96)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)>;
G:=Group( (1,104,70)(2,97,71)(3,98,72)(4,99,65)(5,100,66)(6,101,67)(7,102,68)(8,103,69)(9,82,111)(10,83,112)(11,84,105)(12,85,106)(13,86,107)(14,87,108)(15,88,109)(16,81,110)(17,95,116)(18,96,117)(19,89,118)(20,90,119)(21,91,120)(22,92,113)(23,93,114)(24,94,115)(25,54,75)(26,55,76)(27,56,77)(28,49,78)(29,50,79)(30,51,80)(31,52,73)(32,53,74)(33,47,63)(34,48,64)(35,41,57)(36,42,58)(37,43,59)(38,44,60)(39,45,61)(40,46,62), (1,108,22,42,54)(2,43,109,55,23)(3,56,44,24,110)(4,17,49,111,45)(5,112,18,46,50)(6,47,105,51,19)(7,52,48,20,106)(8,21,53,107,41)(9,61,99,95,78)(10,96,62,79,100)(11,80,89,101,63)(12,102,73,64,90)(13,57,103,91,74)(14,92,58,75,104)(15,76,93,97,59)(16,98,77,60,94)(25,70,87,113,36)(26,114,71,37,88)(27,38,115,81,72)(28,82,39,65,116)(29,66,83,117,40)(30,118,67,33,84)(31,34,119,85,68)(32,86,35,69,120), (1,54)(2,23)(3,110)(4,45)(5,50)(6,19)(7,106)(8,41)(9,95)(10,62)(12,102)(13,91)(14,58)(16,98)(17,111)(20,52)(21,107)(24,56)(25,70)(26,37)(27,115)(29,66)(30,33)(31,119)(35,69)(36,87)(39,65)(40,83)(42,108)(43,55)(46,112)(47,51)(57,103)(59,76)(61,99)(63,80)(67,118)(68,85)(71,114)(72,81)(73,90)(75,104)(77,94)(79,100)(82,116)(86,120)(89,101)(93,97), (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), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(73,77)(75,79)(81,85)(83,87)(90,94)(92,96)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119) );
G=PermutationGroup([(1,104,70),(2,97,71),(3,98,72),(4,99,65),(5,100,66),(6,101,67),(7,102,68),(8,103,69),(9,82,111),(10,83,112),(11,84,105),(12,85,106),(13,86,107),(14,87,108),(15,88,109),(16,81,110),(17,95,116),(18,96,117),(19,89,118),(20,90,119),(21,91,120),(22,92,113),(23,93,114),(24,94,115),(25,54,75),(26,55,76),(27,56,77),(28,49,78),(29,50,79),(30,51,80),(31,52,73),(32,53,74),(33,47,63),(34,48,64),(35,41,57),(36,42,58),(37,43,59),(38,44,60),(39,45,61),(40,46,62)], [(1,108,22,42,54),(2,43,109,55,23),(3,56,44,24,110),(4,17,49,111,45),(5,112,18,46,50),(6,47,105,51,19),(7,52,48,20,106),(8,21,53,107,41),(9,61,99,95,78),(10,96,62,79,100),(11,80,89,101,63),(12,102,73,64,90),(13,57,103,91,74),(14,92,58,75,104),(15,76,93,97,59),(16,98,77,60,94),(25,70,87,113,36),(26,114,71,37,88),(27,38,115,81,72),(28,82,39,65,116),(29,66,83,117,40),(30,118,67,33,84),(31,34,119,85,68),(32,86,35,69,120)], [(1,54),(2,23),(3,110),(4,45),(5,50),(6,19),(7,106),(8,41),(9,95),(10,62),(12,102),(13,91),(14,58),(16,98),(17,111),(20,52),(21,107),(24,56),(25,70),(26,37),(27,115),(29,66),(30,33),(31,119),(35,69),(36,87),(39,65),(40,83),(42,108),(43,55),(46,112),(47,51),(57,103),(59,76),(61,99),(63,80),(67,118),(68,85),(71,114),(72,81),(73,90),(75,104),(77,94),(79,100),(82,116),(86,120),(89,101),(93,97)], [(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)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64),(66,70),(68,72),(73,77),(75,79),(81,85),(83,87),(90,94),(92,96),(98,102),(100,104),(106,110),(108,112),(113,117),(115,119)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 2 | 0 | 0 | 0 | 0 |
| 152 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 0 | 117 | 117 |
| 0 | 0 | 117 | 117 | 0 | 43 |
| 0 | 0 | 124 | 167 | 124 | 0 |
| 0 | 0 | 198 | 74 | 74 | 198 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 240 | 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 | 1 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,240,0,0,1,0,0,240,0,1,0,0,0,240,1,0,0],[240,152,0,0,0,0,2,1,0,0,0,0,0,0,43,117,124,198,0,0,0,117,167,74,0,0,117,0,124,74,0,0,117,43,0,198],[240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 24A | ··· | 24P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 10 | 4 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | M4(2) | C3×M4(2) | F5 | C2×F5 | C2×F5 | C3×F5 | C6×F5 | C6×F5 | D5⋊M4(2) | C3×D5⋊M4(2) |
| kernel | C3×D5⋊M4(2) | C3×D5⋊C8 | C3×C4.F5 | C3×C22.F5 | D5×C2×C12 | D5⋊M4(2) | D5×C12 | C2×C60 | D5×C2×C6 | D5⋊C8 | C4.F5 | C22.F5 | C2×C4×D5 | C4×D5 | C2×C20 | C22×D5 | C3×D5 | D5 | C2×C12 | C12 | C2×C6 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_3\times D_5\rtimes M_{4(2)} % in TeX
G:=Group("C3xD5:M4(2)"); // GroupNames label
G:=SmallGroup(480,1049);
// by ID
G=gap.SmallGroup(480,1049);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,1094,102,9414,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀