direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×C22⋊F5, D5.3(C6×D4), (C22×C6)⋊3F5, C23⋊3(C3×F5), C22⋊3(C6×F5), D10⋊8(C2×C12), (C22×C30)⋊8C4, (C6×D5).82D4, (C6×F5)⋊4C22, (C22×F5)⋊3C6, C30⋊3(C22⋊C4), (C22×C10)⋊6C12, D10.20(C3×D4), (C22×D5)⋊7C12, (C23×D5).5C6, C6.57(C22×F5), C30.95(C22×C4), (C6×D5).72C23, C10.13(C22×C12), D10.13(C22×C6), C5⋊(C6×C22⋊C4), (C2×F5)⋊(C2×C6), (C2×C6×F5)⋊5C2, C10⋊(C3×C22⋊C4), (D5×C2×C6)⋊12C4, D5⋊(C3×C22⋊C4), (C2×C6)⋊7(C2×F5), (C2×C30)⋊9(C2×C4), C2.13(C2×C6×F5), C15⋊7(C2×C22⋊C4), (C2×C10)⋊5(C2×C12), (C6×D5)⋊33(C2×C4), (D5×C22×C6).8C2, (C3×D5).14(C2×D4), (C3×D5)⋊5(C22⋊C4), (D5×C2×C6).152C22, (C22×D5).41(C2×C6), SmallGroup(480,1059)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C22⋊F5
G = < a,b,c,d,e | a6=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 1000 in 264 conjugacy classes, 92 normal (28 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, D5, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C22×C6, C22×C6, C3×D5, C3×D5, C30, C30, C30, C2×C22⋊C4, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C3×C22⋊C4, C22×C12, C23×C6, C3×F5, C6×D5, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C22⋊F5, C22×F5, C23×D5, C6×C22⋊C4, C6×F5, C6×F5, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, C2×C22⋊F5, C3×C22⋊F5, C2×C6×F5, D5×C22×C6, C6×C22⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, F5, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C2×F5, C3×C22⋊C4, C22×C12, C6×D4, C3×F5, C22⋊F5, C22×F5, C6×C22⋊C4, C6×F5, C2×C22⋊F5, C3×C22⋊F5, C2×C6×F5, C6×C22⋊F5
►On 120 points
Generators in S120
(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)
(1 108)(2 103)(3 104)(4 105)(5 106)(6 107)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 96)(26 91)(27 92)(28 93)(29 94)(30 95)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 75)(44 76)(45 77)(46 78)(47 73)(48 74)(49 86)(50 87)(51 88)(52 89)(53 90)(54 85)(67 119)(68 120)(69 115)(70 116)(71 117)(72 118)(79 101)(80 102)(81 97)(82 98)(83 99)(84 100)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 35)(8 36)(9 31)(10 32)(11 33)(12 34)(13 111)(14 112)(15 113)(16 114)(17 109)(18 110)(19 52)(20 53)(21 54)(22 49)(23 50)(24 51)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(55 89)(56 90)(57 85)(58 86)(59 87)(60 88)(61 75)(62 76)(63 77)(64 78)(65 73)(66 74)(67 94)(68 95)(69 96)(70 91)(71 92)(72 93)(79 106)(80 107)(81 108)(82 103)(83 104)(84 105)
(1 25 41 87 109)(2 26 42 88 110)(3 27 37 89 111)(4 28 38 90 112)(5 29 39 85 113)(6 30 40 86 114)(7 108 96 65 50)(8 103 91 66 51)(9 104 92 61 52)(10 105 93 62 53)(11 106 94 63 54)(12 107 95 64 49)(13 99 117 43 55)(14 100 118 44 56)(15 101 119 45 57)(16 102 120 46 58)(17 97 115 47 59)(18 98 116 48 60)(19 31 83 71 75)(20 32 84 72 76)(21 33 79 67 77)(22 34 80 68 78)(23 35 81 69 73)(24 36 82 70 74)
(1 100)(2 101)(3 102)(4 97)(5 98)(6 99)(7 53 96 62)(8 54 91 63)(9 49 92 64)(10 50 93 65)(11 51 94 66)(12 52 95 61)(13 86 117 40)(14 87 118 41)(15 88 119 42)(16 89 120 37)(17 90 115 38)(18 85 116 39)(19 68 75 34)(20 69 76 35)(21 70 77 36)(22 71 78 31)(23 72 73 32)(24 67 74 33)(25 44 109 56)(26 45 110 57)(27 46 111 58)(28 47 112 59)(29 48 113 60)(30 43 114 55)(79 82)(80 83)(81 84)(103 106)(104 107)(105 108)
G:=sub<Sym(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), (1,108)(2,103)(3,104)(4,105)(5,106)(6,107)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,96)(26,91)(27,92)(28,93)(29,94)(30,95)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,75)(44,76)(45,77)(46,78)(47,73)(48,74)(49,86)(50,87)(51,88)(52,89)(53,90)(54,85)(67,119)(68,120)(69,115)(70,116)(71,117)(72,118)(79,101)(80,102)(81,97)(82,98)(83,99)(84,100), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,35)(8,36)(9,31)(10,32)(11,33)(12,34)(13,111)(14,112)(15,113)(16,114)(17,109)(18,110)(19,52)(20,53)(21,54)(22,49)(23,50)(24,51)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(55,89)(56,90)(57,85)(58,86)(59,87)(60,88)(61,75)(62,76)(63,77)(64,78)(65,73)(66,74)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93)(79,106)(80,107)(81,108)(82,103)(83,104)(84,105), (1,25,41,87,109)(2,26,42,88,110)(3,27,37,89,111)(4,28,38,90,112)(5,29,39,85,113)(6,30,40,86,114)(7,108,96,65,50)(8,103,91,66,51)(9,104,92,61,52)(10,105,93,62,53)(11,106,94,63,54)(12,107,95,64,49)(13,99,117,43,55)(14,100,118,44,56)(15,101,119,45,57)(16,102,120,46,58)(17,97,115,47,59)(18,98,116,48,60)(19,31,83,71,75)(20,32,84,72,76)(21,33,79,67,77)(22,34,80,68,78)(23,35,81,69,73)(24,36,82,70,74), (1,100)(2,101)(3,102)(4,97)(5,98)(6,99)(7,53,96,62)(8,54,91,63)(9,49,92,64)(10,50,93,65)(11,51,94,66)(12,52,95,61)(13,86,117,40)(14,87,118,41)(15,88,119,42)(16,89,120,37)(17,90,115,38)(18,85,116,39)(19,68,75,34)(20,69,76,35)(21,70,77,36)(22,71,78,31)(23,72,73,32)(24,67,74,33)(25,44,109,56)(26,45,110,57)(27,46,111,58)(28,47,112,59)(29,48,113,60)(30,43,114,55)(79,82)(80,83)(81,84)(103,106)(104,107)(105,108)>;
G:=Group( (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), (1,108)(2,103)(3,104)(4,105)(5,106)(6,107)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,96)(26,91)(27,92)(28,93)(29,94)(30,95)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,75)(44,76)(45,77)(46,78)(47,73)(48,74)(49,86)(50,87)(51,88)(52,89)(53,90)(54,85)(67,119)(68,120)(69,115)(70,116)(71,117)(72,118)(79,101)(80,102)(81,97)(82,98)(83,99)(84,100), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,35)(8,36)(9,31)(10,32)(11,33)(12,34)(13,111)(14,112)(15,113)(16,114)(17,109)(18,110)(19,52)(20,53)(21,54)(22,49)(23,50)(24,51)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(55,89)(56,90)(57,85)(58,86)(59,87)(60,88)(61,75)(62,76)(63,77)(64,78)(65,73)(66,74)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93)(79,106)(80,107)(81,108)(82,103)(83,104)(84,105), (1,25,41,87,109)(2,26,42,88,110)(3,27,37,89,111)(4,28,38,90,112)(5,29,39,85,113)(6,30,40,86,114)(7,108,96,65,50)(8,103,91,66,51)(9,104,92,61,52)(10,105,93,62,53)(11,106,94,63,54)(12,107,95,64,49)(13,99,117,43,55)(14,100,118,44,56)(15,101,119,45,57)(16,102,120,46,58)(17,97,115,47,59)(18,98,116,48,60)(19,31,83,71,75)(20,32,84,72,76)(21,33,79,67,77)(22,34,80,68,78)(23,35,81,69,73)(24,36,82,70,74), (1,100)(2,101)(3,102)(4,97)(5,98)(6,99)(7,53,96,62)(8,54,91,63)(9,49,92,64)(10,50,93,65)(11,51,94,66)(12,52,95,61)(13,86,117,40)(14,87,118,41)(15,88,119,42)(16,89,120,37)(17,90,115,38)(18,85,116,39)(19,68,75,34)(20,69,76,35)(21,70,77,36)(22,71,78,31)(23,72,73,32)(24,67,74,33)(25,44,109,56)(26,45,110,57)(27,46,111,58)(28,47,112,59)(29,48,113,60)(30,43,114,55)(79,82)(80,83)(81,84)(103,106)(104,107)(105,108) );
G=PermutationGroup([[(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)], [(1,108),(2,103),(3,104),(4,105),(5,106),(6,107),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,96),(26,91),(27,92),(28,93),(29,94),(30,95),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,75),(44,76),(45,77),(46,78),(47,73),(48,74),(49,86),(50,87),(51,88),(52,89),(53,90),(54,85),(67,119),(68,120),(69,115),(70,116),(71,117),(72,118),(79,101),(80,102),(81,97),(82,98),(83,99),(84,100)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,35),(8,36),(9,31),(10,32),(11,33),(12,34),(13,111),(14,112),(15,113),(16,114),(17,109),(18,110),(19,52),(20,53),(21,54),(22,49),(23,50),(24,51),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(55,89),(56,90),(57,85),(58,86),(59,87),(60,88),(61,75),(62,76),(63,77),(64,78),(65,73),(66,74),(67,94),(68,95),(69,96),(70,91),(71,92),(72,93),(79,106),(80,107),(81,108),(82,103),(83,104),(84,105)], [(1,25,41,87,109),(2,26,42,88,110),(3,27,37,89,111),(4,28,38,90,112),(5,29,39,85,113),(6,30,40,86,114),(7,108,96,65,50),(8,103,91,66,51),(9,104,92,61,52),(10,105,93,62,53),(11,106,94,63,54),(12,107,95,64,49),(13,99,117,43,55),(14,100,118,44,56),(15,101,119,45,57),(16,102,120,46,58),(17,97,115,47,59),(18,98,116,48,60),(19,31,83,71,75),(20,32,84,72,76),(21,33,79,67,77),(22,34,80,68,78),(23,35,81,69,73),(24,36,82,70,74)], [(1,100),(2,101),(3,102),(4,97),(5,98),(6,99),(7,53,96,62),(8,54,91,63),(9,49,92,64),(10,50,93,65),(11,51,94,66),(12,52,95,61),(13,86,117,40),(14,87,118,41),(15,88,119,42),(16,89,120,37),(17,90,115,38),(18,85,116,39),(19,68,75,34),(20,69,76,35),(21,70,77,36),(22,71,78,31),(23,72,73,32),(24,67,74,33),(25,44,109,56),(26,45,110,57),(27,46,111,58),(28,47,112,59),(29,48,113,60),(30,43,114,55),(79,82),(80,83),(81,84),(103,106),(104,107),(105,108)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 5 | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 6S | 6T | 6U | 6V | 10A | ··· | 10G | 12A | ··· | 12P | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 1 | 1 | 10 | ··· | 10 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | C3×D4 | F5 | C2×F5 | C3×F5 | C22⋊F5 | C6×F5 | C3×C22⋊F5 |
| kernel | C6×C22⋊F5 | C3×C22⋊F5 | C2×C6×F5 | D5×C22×C6 | C2×C22⋊F5 | D5×C2×C6 | C22×C30 | C22⋊F5 | C22×F5 | C23×D5 | C22×D5 | C22×C10 | C6×D5 | D10 | C22×C6 | C2×C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 6 | 2 | 8 | 4 | 2 | 12 | 4 | 4 | 8 | 1 | 3 | 2 | 4 | 6 | 8 |
Matrix representation of C6×C22⋊F5 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 60 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 0 | 47 | 47 |
| 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 14 | 7 | 14 |
| 0 | 0 | 47 | 47 | 0 | 54 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 | 60 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 50 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,14],[60,0,0,0,0,0,2,1,0,0,0,0,0,0,54,14,0,47,0,0,0,7,14,47,0,0,47,14,7,0,0,0,47,0,14,54],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[50,50,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
C6×C22⋊F5 in GAP, Magma, Sage, TeX
C_6\times C_2^2\rtimes F_5
% in TeX
G:=Group("C6xC2^2:F5"); // GroupNames label
G:=SmallGroup(480,1059);
// by ID
G=gap.SmallGroup(480,1059);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,1094,9414,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations