direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23⋊F5, C23⋊(C3×F5), C22⋊F5⋊2C6, (C22×C6)⋊1F5, C15⋊6(C23⋊C4), (C22×C30)⋊7C4, (C6×Dic5)⋊4C4, D10.4(C3×D4), (C6×D5).39D4, (C22×C10)⋊5C12, C22.4(C6×F5), (C2×Dic5)⋊2C12, C6.35(C22⋊F5), C30.35(C22⋊C4), C5⋊2(C3×C23⋊C4), (C2×C5⋊D4).7C6, (C3×C22⋊F5)⋊6C2, (C2×C6).28(C2×F5), (C2×C30).53(C2×C4), (C6×C5⋊D4).16C2, C10.9(C3×C22⋊C4), (D5×C2×C6).90C22, (C2×C10).10(C2×C12), C2.10(C3×C22⋊F5), (C22×D5).17(C2×C6), SmallGroup(480,291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23⋊F5
G = < a,b,c,d,e,f | a3=b2=c2=d2=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >
Subgroups: 472 in 104 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, F5, D10, D10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C23⋊C4, C2×Dic5, C5⋊D4, C2×F5, C22×D5, C22×C10, C3×C22⋊C4, C6×D4, C3×Dic5, C3×F5, C6×D5, C6×D5, C2×C30, C2×C30, C22⋊F5, C2×C5⋊D4, C3×C23⋊C4, C6×Dic5, C3×C5⋊D4, C6×F5, D5×C2×C6, C22×C30, C23⋊F5, C3×C22⋊F5, C6×C5⋊D4, C3×C23⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C23⋊C4, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C23⋊C4, C6×F5, C23⋊F5, C3×C22⋊F5, C3×C23⋊F5
►On 120 points
Generators in S120
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 64)(2 65)(3 61)(4 62)(5 63)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)(111 116)(112 117)(113 118)(114 119)(115 120)
(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 64 14 79)(2 61 13 77)(3 63 12 80)(4 65 11 78)(5 62 15 76)(6 68 17 75)(7 70 16 73)(8 67 20 71)(9 69 19 74)(10 66 18 72)(21 83 32 100)(22 85 31 98)(23 82 35 96)(24 84 34 99)(25 81 33 97)(26 88 37 95)(27 90 36 93)(28 87 40 91)(29 89 39 94)(30 86 38 92)(41 103 52 120)(42 105 51 118)(43 102 55 116)(44 104 54 119)(45 101 53 117)(46 108 57 115)(47 110 56 113)(48 107 60 111)(49 109 59 114)(50 106 58 112)
G:=sub<Sym(120)| (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,64)(2,65)(3,61)(4,62)(5,63)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120), (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), (1,64,14,79)(2,61,13,77)(3,63,12,80)(4,65,11,78)(5,62,15,76)(6,68,17,75)(7,70,16,73)(8,67,20,71)(9,69,19,74)(10,66,18,72)(21,83,32,100)(22,85,31,98)(23,82,35,96)(24,84,34,99)(25,81,33,97)(26,88,37,95)(27,90,36,93)(28,87,40,91)(29,89,39,94)(30,86,38,92)(41,103,52,120)(42,105,51,118)(43,102,55,116)(44,104,54,119)(45,101,53,117)(46,108,57,115)(47,110,56,113)(48,107,60,111)(49,109,59,114)(50,106,58,112)>;
G:=Group( (1,44,24)(2,45,25)(3,41,21)(4,42,22)(5,43,23)(6,46,26)(7,47,27)(8,48,28)(9,49,29)(10,50,30)(11,51,31)(12,52,32)(13,53,33)(14,54,34)(15,55,35)(16,56,36)(17,57,37)(18,58,38)(19,59,39)(20,60,40)(61,101,81)(62,102,82)(63,103,83)(64,104,84)(65,105,85)(66,106,86)(67,107,87)(68,108,88)(69,109,89)(70,110,90)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (1,64)(2,65)(3,61)(4,62)(5,63)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110)(111,116)(112,117)(113,118)(114,119)(115,120), (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), (1,64,14,79)(2,61,13,77)(3,63,12,80)(4,65,11,78)(5,62,15,76)(6,68,17,75)(7,70,16,73)(8,67,20,71)(9,69,19,74)(10,66,18,72)(21,83,32,100)(22,85,31,98)(23,82,35,96)(24,84,34,99)(25,81,33,97)(26,88,37,95)(27,90,36,93)(28,87,40,91)(29,89,39,94)(30,86,38,92)(41,103,52,120)(42,105,51,118)(43,102,55,116)(44,104,54,119)(45,101,53,117)(46,108,57,115)(47,110,56,113)(48,107,60,111)(49,109,59,114)(50,106,58,112) );
G=PermutationGroup([[(1,44,24),(2,45,25),(3,41,21),(4,42,22),(5,43,23),(6,46,26),(7,47,27),(8,48,28),(9,49,29),(10,50,30),(11,51,31),(12,52,32),(13,53,33),(14,54,34),(15,55,35),(16,56,36),(17,57,37),(18,58,38),(19,59,39),(20,60,40),(61,101,81),(62,102,82),(63,103,83),(64,104,84),(65,105,85),(66,106,86),(67,107,87),(68,108,88),(69,109,89),(70,110,90),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(1,64),(2,65),(3,61),(4,62),(5,63),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110),(111,116),(112,117),(113,118),(114,119),(115,120)], [(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)], [(1,64,14,79),(2,61,13,77),(3,63,12,80),(4,65,11,78),(5,62,15,76),(6,68,17,75),(7,70,16,73),(8,67,20,71),(9,69,19,74),(10,66,18,72),(21,83,32,100),(22,85,31,98),(23,82,35,96),(24,84,34,99),(25,81,33,97),(26,88,37,95),(27,90,36,93),(28,87,40,91),(29,89,39,94),(30,86,38,92),(41,103,52,120),(42,105,51,118),(43,102,55,116),(44,104,54,119),(45,101,53,117),(46,108,57,115),(47,110,56,113),(48,107,60,111),(49,109,59,114),(50,106,58,112)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4E | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10G | 12A | ··· | 12J | 15A | 15B | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 1 | 1 | 20 | ··· | 20 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 20 | ··· | 20 | 4 | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | F5 | C23⋊C4 | C2×F5 | C3×F5 | C22⋊F5 | C3×C23⋊C4 | C6×F5 | C23⋊F5 | C3×C22⋊F5 | C3×C23⋊F5 |
| kernel | C3×C23⋊F5 | C3×C22⋊F5 | C6×C5⋊D4 | C23⋊F5 | C6×Dic5 | C22×C30 | C22⋊F5 | C2×C5⋊D4 | C2×Dic5 | C22×C10 | C6×D5 | D10 | C22×C6 | C15 | C2×C6 | C23 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C23⋊F5 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 27 | 58 | 13 | 58 |
| 54 | 24 | 10 | 10 |
| 51 | 51 | 37 | 7 |
| 3 | 48 | 3 | 34 |
| 54 | 14 | 0 | 47 |
| 0 | 7 | 14 | 47 |
| 47 | 14 | 7 | 0 |
| 47 | 0 | 14 | 54 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 1 | 0 | 0 |
| 60 | 0 | 1 | 0 |
| 60 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 27 | 13 | 27 | 58 |
| 54 | 10 | 24 | 24 |
| 51 | 37 | 37 | 51 |
| 3 | 3 | 34 | 48 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[27,54,51,3,58,24,51,48,13,10,37,3,58,10,7,34],[54,0,47,47,14,7,14,0,0,14,7,14,47,47,0,54],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,60,60,60,1,0,0,0,0,1,0,0,0,0,1,0],[27,54,51,3,13,10,37,3,27,24,37,34,58,24,51,48] >;
C3×C23⋊F5 in GAP, Magma, Sage, TeX
C_3\times C_2^3\rtimes F_5
% in TeX
G:=Group("C3xC2^3:F5"); // GroupNames label
G:=SmallGroup(480,291);
// by ID
G=gap.SmallGroup(480,291);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,850,2524,9414,1595]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^5=f^4=1,a*b=b*a,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,f*b*f^-1=b*c*d,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